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A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3
.Chaos theory is a field of study in mathematics, physics, and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions.^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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^ Chaos, then, appears to be unstable aperiodic behavior in nonlinear dynamical systems.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Hamiltonian chaos, then, is chaotic behavior in a Hamiltonian system.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.This sensitivity is popularly referred to as the butterfly effect.^ One of the first characteristics applicable for warfare is extreme sensitivity to initial conditions, also known as the "butterfly effect."

.Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general.^ This makes the long-term evolution of the system impossible, in principle, to predict.
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^ This is unlikely to work, and impossible to work in the long term.
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^ Only a small change in the initial conditions can drastically change the long-term behavior of a system.
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[1] .This happens even though these systems are deterministic, meaning that their future behaviour is fully determined by their initial conditions, with no random elements involved.^ What do these teams mean though?
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^ These means involve the essential questions that drive recognition of both the problems and prospects for change in work processes.
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^ Because of SDIC, nonlinear chaotic systems whose initial states can be located only within a small neighborhood ε of state space will have future states that can be located only within a much larger patch δ.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

[2] .In other words, the deterministic nature of these systems does not make them predictable.^ Simple deterministic systems with only a few elements can generate random behavior, and that randomness is fundamental; gathering more information does not make it disappear.
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^ In other words, to predict the future state of a system with certainty, you need to know the initial conditions with infinite accuracy, since errors increase rapidly with even the slightest inaccuracy.
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^ These, and other factors mentioned by Koehler and Comfort below, may condition how quickly a disaster wide response system emerges.
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[3] .This behavior is known as deterministic chaos, or simply chaos.^ For instance, Kellert characterizes chaos theory as “the qualitative study of unstable aperiodic behavior in deterministic nonlinear systems” (Kellert 1993, p.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Stephen Kellert defines chaos theory as “the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems” (1993, p.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Kellert (1993) defines chaos theory as "the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems" (p.
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.Chaotic behavior can be observed in many natural systems, such as the weather.^ Hamiltonian chaos, then, is chaotic behavior in a Hamiltonian system.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Phenomenologically, the kinds of chaotic behavior we see in real-world systems exhibit features such as SDIC, aperiodicity, unpredictability, instability under small perturbations and apparent randomness.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Low level chaotic behavior may simply be evidence of a system's adaptive response to its environment.
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[4] .Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.^ Perhaps the fractal character of strange attractors is an artifact introduced through the various idealizations and approximations used to derive such chaotic models.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Questions about realism and explanation in chaotic dynamics ( §5 ) are relevant here as well as the faithful model assumption.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ These assumptions allow us to develop mathematical models for the evolution of these points in state space and such models are taken to represent (perhaps through an isomorphism or some more complicated relation) the physical systems of interest.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

Contents

Applications

.Chaos theory is applied in many scientific disciplines: mathematics, programming, microbiology, biology, computer science, economics,[5][6][7] engineering,[8] finance,[9][10] philosophy, physics, politics, population dynamics, psychology, and robotics.^ Unfortunately, this would mean that chaos theory would be only a mathematical theory and not a physical one.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ To accomplish this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.
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^ It turns out that in mathematical theory the change from order and predictability into unpredictability or chaos for dynamic systems is governed by a single law, and that the 'route' between the two conditions is a universal one.
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[11]
.Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical devices, as well as computer models of chaotic processes.^ Hamiltonian chaos, then, is chaotic behavior in a Hamiltonian system.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Computer simulations well be necessary to further validate these models.
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^ However, in nonlinear contexts, where one might be constructing a model from a data set generated by observing a system, there are potentially many nonlinear models that can be constructed, where each model is as empirically adequate to the system behavior as any other.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.Observations of chaotic behavior in nature include changes in weather,[4] the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations.^ The weather is another system that changes dramatically over time.
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^ Hamiltonian chaos, then, is chaotic behavior in a Hamiltonian system.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ A dynamical system is any process that moves or changes in time.
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.There is some controversy over the existence of chaotic dynamics in plate tectonics and in economics.^ If there is a breakdown of determinism in chaotic systems, that can only occur if there is some kind of indeterminism introduced such that the property of unique evolution is rendered false (e.g., §4 below).
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ There are no axioms—no laws—no deductive structures, no linking of observational statements to theoretical statements whatsoever in the literature on chaotic dynamics.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ But it would also be the case that there is nothing special about such explanations: there are processes and interactions that cause the dynamics to have chaotic properties.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

[12][13][14]
.One of the most successful applications of chaos theory has been in ecology, where dynamical systems such as the Ricker model have been used to show how population growth under density dependence can lead to chaotic dynamics.^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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^ "How does one start using chaos theory to achieve a desired outcome?
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^ RE: Using the chaos theory .
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.Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.^ Applied chaos theory: A paradigm for complexity .
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^ Chaos theory studies systems that are flowing.
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^ A chaos scientist does not look at the end product but takes a look instead at the basic principles and history that make up this seemingly random action.
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[15]
.A related field of physics called quantum chaos theory investigates the relationship between chaos and quantum mechanics.^ Furthermore, the relationship between the state spaces of chaotic models and the spaces of idealized physical systems is quite delicate, which seems to be a dissimilarity between classical mechanics and “chaos theory”.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Unfortunately, this would mean that chaos theory would be only a mathematical theory and not a physical one.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ This terribly thin demarcation line is suggestive of the edge of chaos which is being investigated today in complexity theory.
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.The correspondence principle states that classical mechanics is a special case of quantum mechanics, the classical limit.^ Premise (B) expresses the precision limit for the state of minimum uncertainty for measuring momentum and position pairs in an N -dimensional quantum system (note, the exponent is 2 N in the case of measuring uncorrelated electrons).
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ There is a serious open question as to whether the indeterminism in quantum mechanics is simply the result of ignorance due to epistemic limitations or if it is an ontological feature of the quantum world.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ But it is neither clear how this would work in the case of nonlinear systems in classical mechanics, nor how this would work for chaotic models in biology, economics and other disciplines.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, it is unclear how exponential sensitivity to initial conditions can arise in practice in classical chaos.^ SD arguments purport to demonstrate that chaos in classical systems can amplify quantum fluctuations due to sensitivity to the smallest changes in initial conditions.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ He suggests that exponential instability—the exponential divergence of two trajectories issuing forth from neighboring initial conditions—is a necessary condition, but leaves it open as to whether it is sufficient.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Design and program a PC based disaster simulation program that could be used to investigate the sensitivity to initial conditions of key components of the response for various types of disasters.
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[16] .Recently, another field, called relativistic chaos,[17] has emerged to describe systems that follow the laws of general relativity.^ Put another way, chaos is a type of non-linear behavior emerging along a universal route.
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^ In full blown chaos, disorder almost displaces order; a system may end up almost any place in an outcome field.
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^ A further question regarding chaos and realism is the following: Is chaos only a feature of our mathematical models or is it a genuine feature of actual systems in our world?
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.The initial conditions of three or more bodies interacting through gravitational attraction (see the n-body problem) can be arranged to produce chaotic motion.^ As per the current mathematical theory a chaotic system is defined as showing "sensitivity to initial conditions".
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^ It may be that sensitivity to initial conditions and fortunate relationships with other organizations have more to do with a successful response than disaster management skill.
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^ The system had only three equations this time and it no longer had anything to do with the convection, but it did have sensitive dependence on its initial conditions.
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Chaotic dynamics

.
The map defined by x → 4 x (1 – x) and yx + y if x + y < 1 (x + y – 1 otherwise) displays sensitivity to initial conditions.
^ Design and program a PC based disaster simulation program that could be used to investigate the sensitivity to initial conditions of key components of the response for various types of disasters.
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^ It may be that sensitivity to initial conditions and fortunate relationships with other organizations have more to do with a successful response than disaster management skill.
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^ The system had only three equations this time and it no longer had anything to do with the convection, but it did have sensitive dependence on its initial conditions.
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Here two series of x and y values diverge markedly over time from a tiny initial difference.
.In common usage, "chaos" means "a state of disorder",[18] but the adjective "chaotic" is defined more precisely in chaos theory.^ To apply Chaos Theory, a single measured variable x(n) = x(t0 + nt) with a starting time, t0, and a lead time, t, provides an n-dimensional space, or phase space, that represents the full multivariate state space of the system; up to 4 dimensions may be required to represent the phase space for a chaotic system.
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^ Unfortunately, this would mean that chaos theory would be only a mathematical theory and not a physical one.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ But, believing in Chaos Theory doesn't mean you have to surrender to it.
  • Tom Clancy's Splinter Cell: Chaos Theory - Wikiquote 19 January 2010 9:53 UTC en.wikiquote.org [Source type: Original source]

Although there is no universally accepted mathematical definition of chaos, a commonly-used definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:[19]
  1. it must be sensitive to initial conditions,
  2. it must be topologically mixing, and
  3. its periodic orbits must be dense.

Sensitivity to initial conditions

.Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories.^ All chaotic systems seem to have an unusual sensitivity to initial conditions.

^ A strange attractor results if a system is sensitive to initial conditions and is not conservative.

^ The universe's tendency to play dice and the resulting inherent randomness means each initial starting point is arbitrarily closely approximated by other points with significantly different trajectories and outcomes.
  • Chaos Theory and Prototype Brewing 19 January 2010 9:53 UTC www.brewdog.com [Source type: General]

.Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour.^ The universe's tendency to play dice and the resulting inherent randomness means each initial starting point is arbitrarily closely approximated by other points with significantly different trajectories and outcomes.
  • Chaos Theory and Prototype Brewing 19 January 2010 9:53 UTC www.brewdog.com [Source type: General]

^ Different groups and individuals within the organization may be either future or past oriented making it difficult to coordinate to achieve common goals.
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^ Aperiodic behavior never repeats and it continues to manifest the effects of any small perturbation; hence, any prediction of a future state in a given system that is aperiodic is impossible.

.However, it has been shown that the last two properties in the list above actually imply sensitivity to initial conditions[20][21] and if attention is restricted to intervals, the second property implies the other two[22] (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list[23]).^ Is there only one definition for chaos, and if so, is it only a mathematical property or also a physical one?
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Batterman does not actually specify an alternative definition of chaos.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ First, it can be proven that Chaos h implies Chaos d .
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.It is interesting that the most practically significant condition, that of sensitivity to initial conditions, is actually redundant in the definition, being implied by two (or for intervals, one) purely topological conditions, which are therefore of greater interest to mathematicians.^ He suggests that exponential instability—the exponential divergence of two trajectories issuing forth from neighboring initial conditions—is a necessary condition, but leaves it open as to whether it is sufficient.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Design and program a PC based disaster simulation program that could be used to investigate the sensitivity to initial conditions of key components of the response for various types of disasters.
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^ It may be that sensitivity to initial conditions and fortunate relationships with other organizations have more to do with a successful response than disaster management skill.
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.Sensitivity to initial conditions is popularly known as the "butterfly effect," so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C. entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena.^ The Butterfly Effect reflects how changes on the small scale, can influence things on the large scale.
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^ In nonlinear systems small changes or small errors can have big effects.
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^ The butterfly effect is what this effect came to be known as.
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.Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different (even the evolution of simple discrete systems, such as cellular automata, can heavily depend on initial conditions, and Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30).^ Let x (0) and y (0) be initial conditions for two different trajectories.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ A single butterfly flapping its wings today produces a tiny change in the state of the atmosphere.
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^ He suggests that exponential instability—the exponential divergence of two trajectories issuing forth from neighboring initial conditions—is a necessary condition, but leaves it open as to whether it is sufficient.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.A consequence of sensitivity to initial conditions is that if we start with only a finite amount of information about the system (as is usually the case in practice), then beyond a certain time the system will no longer be predictable.^ The system had only three equations this time and it no longer had anything to do with the convection, but it did have sensitive dependence on its initial conditions.
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^ Forecasting Response Bifurcation Points And The Emergence Of Large Scale Disaster Response Systems We have shown that we cannot predict what the disaster response structure will look like no matter how much information we collect.
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^ The conditions of the buckets filling and emptying will no longer be so synchronous as to facilitate just simple rotation; chaos has taken over.
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.This is most familiar in the case of weather, which is generally predictable only about a week ahead.^ This approach of aggregating cases across disasters and disciplines has been used to arrive at generalized findings about disaster management.
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^ That not only ruined any hope of forecasting weather beyond a week or so, but similarly hampered our ability to foresee climate change.
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^ In 1956, Thompson estimated that because of the way small irregularities got magnified as a computation went forward, it would never be possible to compute an accurate prediction of weather more than about two weeks ahead.
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[24]
.The Lyapunov exponent characterises the extent of the sensitivity to initial conditions.^ Design and program a PC based disaster simulation program that could be used to investigate the sensitivity to initial conditions of key components of the response for various types of disasters.
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^ It may be that sensitivity to initial conditions and fortunate relationships with other organizations have more to do with a successful response than disaster management skill.
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^ The system had only three equations this time and it no longer had anything to do with the convection, but it did have sensitive dependence on its initial conditions.
  • Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]

Quantitatively, two trajectories in phase space with initial separation \delta \mathbf{Z}_0 diverge
 | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |
where λ is the Lyapunov exponent. The rate of separation can be different for different orientations of the initial separation vector. .Thus, there is a whole spectrum of Lyapunov exponents — the number of them is equal to the number of dimensions of the phase space.^ This strange phenomena was said to reside in what they called phase space and a whole new element of chaos theory was born.
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^ But in the long run, some of these trajectories could effectively diverge as if there was on-average exponential growth in uncertainties as characterized by global Lyapunov exponents.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ After four bifurcations, the number of attractors possible for similar members of a set explode to fill any phase-space available to it (Region D).
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It is common to just refer to the largest one, i.e. to the Maximal Lyapunov exponent (MLE), because it determines the overall predictability of the system. .A positive MLE is usually taken as an indication that the system is chaotic.^ This is a different version of the faithful model assumption in that now the topological/geometric features of target systems are taken to be faithfully represented by our chaotic models.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Taken together, these three conditions represent an attempt to precisely characterize the kind of irregular, aperiodic behavior we expect chaotic systems to exhibit.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

Topological mixing

The map defined by x → 4 x (1 – x) and yx + y if x + y < 1 (x + y – 1 otherwise) also displays topological mixing. Here the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of points scattered across the space.
.Topological mixing (or topological transitivity) means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region.^ To apply Chaos Theory, a single measured variable x(n) = x(t0 + nt) with a starting time, t0, and a lead time, t, provides an n-dimensional space, or phase space, that represents the full multivariate state space of the system; up to 4 dimensions may be required to represent the phase space for a chaotic system.
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^ To summarize, the folklore that trajectories issuing forth from neighboring points will diverge on-average exponentially in a chaotic region of state space is false in any sense other than for infinitesimal uncertainties in the infinite time limit.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Simple systems such as a pendulum have a point to which they are attracted; more complex systems such as a thermostat are limited to a slightly larger region in phase-space.
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.This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.^ A concept involved in chaotic systems is fractals.
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^ It is a mathematically constructed conceptual space where each dimension corresponds to one variable of the system (Kellert, 1993).
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^ An example in the world of physics of such a system would be a fluid in turbulent motion (1).
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.Topological mixing is often omitted from popular accounts of chaos, which equate chaos with sensitivity to initial conditions.^ In the case of a disaster, the field of action is influenced by the initial conditions that an organization finds itself in following the often abrupt on-set of the disaster.
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^ A complete specification of the initial state of such equations is referred to as the initial conditions for the model, while a characterization of the boundaries for the model domain are known as the boundary conditions.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Design and program a PC based disaster simulation program that could be used to investigate the sensitivity to initial conditions of key components of the response for various types of disasters.
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.However, sensitive dependence on initial conditions alone does not give chaos.^ This phenomenon is also known as sensitive dependence on initial conditions and is common to the chaos theory.
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^ As per the current mathematical theory a chaotic system is defined as showing "sensitivity to initial conditions".
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^ Design and program a PC based disaster simulation program that could be used to investigate the sensitivity to initial conditions of key components of the response for various types of disasters.
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.For example, consider the simple dynamical system produced by repeatedly doubling an initial value.^ If you look at some of the examples giving of dynamical systems, it is clear that some of them are predictable.
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^ One tenet of nonlinear dynamics is that complex systems defy simple formulation and thus may preclude the development of precise mathematical algorithms.
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^ Second, if simple systems can generate complex behavior then imagine what may result when considering the complex organizations and environments that disaster and emergency services managers attempt to handle.
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.This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated.^ Another system in which sensitive dependence on initial conditions is evident is the flip of a coin.
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^ It was a theorem on the sensitive dependence on initial conditions about the frictionless motion of a point on a surface.
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^ More exactly, chaotic dynamical systems are characterized by "sensitive dependence on initial conditions".
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However, this example has no topological mixing, and therefore has no chaos. .Indeed, it has extremely simple behaviour: all points except 0 tend to infinity.^ Repeated application of a simple function causes some of these points to flee toward infinity, while others never wander far from the origin.
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^ J ( x (0)) − J ( y (0))| e λ t ], where the “almost all” caveat is understood as applying for all points in state space except a set of measure zero.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

Density of periodic orbits

.Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.^ More to the point, the definition seems counterintuitive in that it emphasizes periodic orbits rather than aperiodicity, but the latter seems a much better characterization of chaos.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ This is because the cycle repeats itself every two time periods or every two data points; the cycle stabilizes at about point 20.
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^ (Chaos d ) A continuous map f is chaotic if f has an invariant set K ⊆ S such that f satisfies WSD on K , The set of points initiating periodic orbits are dense in K , and f is topologically transitive on K .
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.Topologically mixing systems failing this condition may not display sensitivity to initial conditions, and hence may not be chaotic.^ More exactly, chaotic dynamical systems are characterized by "sensitive dependence on initial conditions".
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^ Sensitivity to Initial Conditions Systems functioning in chaotic regimes may show a tendency to be highly sensitive to their initial conditions.
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^ This established that the system is inherently unpredictable and as a result any slight error in the initial conditions would be amplified rapidly.
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.For example, an irrational rotation of the circle is topologically transitive, but does not have dense periodic orbits, and hence does not have sensitive dependence on initial conditions.^ Design and program a PC based disaster simulation program that could be used to investigate the sensitivity to initial conditions of key components of the response for various types of disasters.
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^ It may be that sensitivity to initial conditions and fortunate relationships with other organizations have more to do with a successful response than disaster management skill.
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^ The system had only three equations this time and it no longer had anything to do with the convection, but it did have sensitive dependence on its initial conditions.
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[25] .The one-dimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits.^ For example, if one begins with a continuous system, by using a Poincaré surface of section—roughly, a two-dimensional plane is defined and one plots the intersections of trajectories with this plane—a discrete map can be generated.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ In the case of one-dimensional maps, however, it can be shown that Chaos h implies Chaos te .
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Is a realist construal of faithfulness threatened by the mapping between models and systems potentially being one-to-many or many-to-many?
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

For example, 0.3454915 → 0.9045085 → 0.3454915 is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem).[26]
.Sharkovskii's theorem is the basis of the Li and Yorke[27] (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.^ For example, it turns out that the ratio of the length of two succeeding bifurcation branches is a universal constant that is applicable to all systems that exhibit this behavior.
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^ Unstable systems are those exhibiting SDIC. Aperiodic behavior means that the system variables never repeat values in any regular fashion.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ For systems exhibiting SDIC, trajectories starting out in a highly localized region of state space will diverge on-average exponentially fast from one another.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

Strange Attractors

The Lorenz attractor displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.
.Some dynamical systems, like the one-dimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere, but in many cases chaotic behaviour is found only in a subset of phase space.^ Chaos research maps the geometry of system dynamics in phase-space.
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^ It will also depend on the complexity of the dynamic that defines the system.
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^ To apply Chaos Theory, a single measured variable x(n) = x(t0 + nt) with a starting time, t0, and a lead time, t, provides an n-dimensional space, or phase space, that represents the full multivariate state space of the system; up to 4 dimensions may be required to represent the phase space for a chaotic system.
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.The cases of most interest arise when the chaotic behaviour takes place on an attractor, since then a large set of initial conditions will lead to orbits that converge to this chaotic region.^ Those points in convergent regions form a Fatou set.
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^ The idea here is that if a model is faithful in reproducing the behavior of the target system to some degree, refining the precision of the initial data fed to the model will lead to its behavior monotonically converging to the target system's behavior.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ An important geometric implication of self-similarity is that there is no inherent size scale so that we can take as large a magnification of as small a region of the attractor as we want and a statistically similarly structure will be repeated (Hilborn 1994, p.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit.^ A Julia set is a set of extraordinary points that separate different basins of attraction.
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^ A set of points whose forward orbits move toward the same limit point is called a basin of attraction , and this limit point is known as the set's attractor .
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^ One plots the boundary between basins of attraction; the other graphs the attractor of the inverse relation.
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.Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor.^ A value for c that lies on one of the filaments creates a similarly shaped Julia set, because infinity is the only attractor.
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^ Right now we would like to pause along the route to chaos to offer a mock application of such phase transitions for heuristic purposes.
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^ The attractor in figure 5 is derived from a chaotic time series generated by the logistic equation.
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.This attractor results from a simple three-dimensional model of the Lorenz weather system.^ Second, if simple systems can generate complex behavior then imagine what may result when considering the complex organizations and environments that disaster and emergency services managers attempt to handle.
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^ On the other hand, the dissipative chaotic models used to characterize some real-world systems all exhibit strange attractors with fractal geometries.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ When we apply this generalization of dimensionality to the geometric structure of strange attractors, we find that we get noninteger results.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it was not only one of the first, but it is also one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly.^ Attractors - The Order in Chaos One of the interesting qualities of nonlinear dynamics as a paradigm for emergency management is that even when the data we examine look erratic and chaotic we can find a deeper order in the data.
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^ Most have long known that the solar system does not "run with the precision of a Swiss watch."
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^ The first such attractor was identified by Edward Lorenz in his meteorology simulations in the early 1960s (Lorenz, 1963).
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.Unlike fixed-point attractors and limit cycles, the attractors which arise from chaotic systems, known as strange attractors, have great detail and complexity.^ At this point, laminar flow will transition into turbulence (in chaos theory, turbulence is a strange attractor which draws the orderly flowing water into chaos as velocity increases).
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^ Attractor, Strange: A strange attractor is simply the pattern, in visual form, produced by graphing the behavior of a nonlinear system.
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^ Some attractors are called 'strange' attractors since a system behaves in ways not expected by Newtonian physics, propositional logic, rational numbering systems or euclidean geometry.
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.Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map).^ There are two main types of dynamic systems: discrete and continuous.
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^ The first such attractor was identified by Edward Lorenz in his meteorology simulations in the early 1960s (Lorenz, 1963).
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^ If there is a breakdown of determinism in chaotic systems, that can only occur if there is some kind of indeterminism introduced such that the property of unique evolution is rendered false (e.g., §4 below).
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points – Julia sets can be thought of as strange repellers.^ Systems with dissipative structures are called dissipative systems.
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^ A system is said to be 'attracted' to that region, hence the pattern of nonlinear dynamics seen in such a basin is called an attractor.
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^ On the other hand, percolation theory and Comfort's work on communications structures suggest that there are deep rules that may contribute to the emergence of large scale systems.
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.Both strange attractors and Julia sets typically have a fractal structure, and a fractal dimension can be calculated for them.^ In other words strange attractors for chaotic models have an infinite number of layers of repetitive structure .
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Perhaps the fractal character of strange attractors is an artifact introduced through the various idealizations and approximations used to derive such chaotic models.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Strange attractors also are often characterized as possess noninteger or fractal dimension (though not all strange attractors have such dimensionality).
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

Minimum complexity of a chaotic system

.
Bifurcation diagram of the logistic map xr x (1 – x).
^ Using The Logistic Map To Track Disaster Organizational Morphogenesis Recall that the organizational states depicted by the logistic diagram take place during a twenty-four hour period.
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^ According to the logistic diagram, sensitivity to initial conditions increases as an organization moves up to and through a bifurcation point and on to the edge of chaos.
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^ Generally, the map of disaster organization morphogenesis defines where an organization or response system is on the logistic diagram.
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Each vertical slice shows the attractor for a specific value of r. The diagram displays period-doubling as r increases, eventually producing chaos.
.Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality.^ For example, if one begins with a continuous system, by using a Poincaré surface of section—roughly, a two-dimensional plane is defined and one plots the intersections of trajectories with this plane—a discrete map can be generated.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Phenomenologically, the kinds of chaotic behavior we see in real-world systems exhibit features such as SDIC, aperiodicity, unpredictability, instability under small perturbations and apparent randomness.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Would this be evidence that there is a strange attractor in the target system's behavior?
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.However, the Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system (specified by differential equations) if it has three or more dimensions.^ Would this be evidence that there is a strange attractor in the target system's behavior?
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ A system is said to be 'attracted' to that region, hence the pattern of nonlinear dynamics seen in such a basin is called an attractor.
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^ There is much more that will be learned over the next several decades about complex systems and nonlinear dynamics as scholars and managers understand more about this new vision of reality.
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.Finite dimensional linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be either nonlinear, or infinite-dimensional.^ Because of SDIC, nonlinear chaotic systems whose initial states can be located only within a small neighborhood ε of state space will have future states that can be located only within a much larger patch δ.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ And one has to investigate the local finite-time dynamics for each system because these may not yield any on-average growth in uncertainties (e.g., Smith, Ziehmann, Fraedrich 1999).
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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.The Poincaré–Bendixson theorem states that a two dimensional differential equation has very regular behavior.^ Which means that if you very slightly change a parameter in an equation or system, very different behaviors can result (29).
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^ When we see two branches this just means that the long term behavior of the system is now alternating between two different states, a lower one and an upper one.
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^ Before turning to a more detailed discussion of these three states, does the logistic equation when applied to disaster response data fit this pattern of behavior ?
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.The Lorenz attractor discussed above is generated by a system of three differential equations with a total of seven terms on the right hand side, five of which are linear terms and two of which are quadratic (and therefore nonlinear).^ A system is said to be 'attracted' to that region, hence the pattern of nonlinear dynamics seen in such a basin is called an attractor.
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^ When we see two branches this just means that the long term behavior of the system is now alternating between two different states, a lower one and an upper one.
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^ When a system is mismatched to its environment, there are three generic solutions by which rematch can be obtained: 1) change the system, 2) change the environment or 3) recourse to a third system by which to bridge the two (Young, 1969, 1977).
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.Another well-known chaotic attractor is generated by the Rossler equations with seven terms on the right hand side, only one of which is (quadratic) nonlinear.^ Attractors - The Order in Chaos One of the interesting qualities of nonlinear dynamics as a paradigm for emergency management is that even when the data we examine look erratic and chaotic we can find a deeper order in the data.
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^ Nonlinear equations have been around for a long time, but no one was able to solve them, and traditional scientists and engineers simply ignored all nonlinear portions of their calculations.
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^ Because of SDIC, nonlinear chaotic systems whose initial states can be located only within a small neighborhood ε of state space will have future states that can be located only within a much larger patch δ.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.Sprott[28] found a three dimensional system with just five terms on the right hand side, and with just one quadratic nonlinearity, which exhibits chaos for certain parameter values.^ Nonlinear systems exhibit three distinct types of behavior over time.
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^ Chaos is one possible result of the dynamics of nonlinear systems.
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^ One method for controlling chaos is to alter the parameters of the system.
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.Zhang and Heidel[29][30] showed that, at least for dissipative and conservative quadratic systems, three dimensional quadratic systems with only three or four terms on the right hand side cannot exhibit chaotic behavior.^ Hamiltonian chaos, then, is chaotic behavior in a Hamiltonian system.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Nonlinear systems exhibit three distinct types of behavior over time.
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^ Not only is our body a dissipative system, but our ego as well.
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.The reason is, simply put, that solutions to such systems are asymptotic to a two dimensional surface and therefore solutions are well behaved.^ For example, if one begins with a continuous system, by using a Poincaré surface of section—roughly, a two-dimensional plane is defined and one plots the intersections of trajectories with this plane—a discrete map can be generated.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ When a system is mismatched to its environment, there are three generic solutions by which rematch can be obtained: 1) change the system, 2) change the environment or 3) recourse to a third system by which to bridge the two (Young, 1969, 1977).
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^ Such efforts are unwise for two reasons; first, the costs of control efforts would affect profits (or wages if it were a worker owned and operated firm).
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.While the Poincaré–Bendixson theorem means that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behaviour.^ Dynamical systems are comprised of two elements.
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^ For example, if one begins with a continuous system, by using a Poincaré surface of section—roughly, a two-dimensional plane is defined and one plots the intersections of trajectories with this plane—a discrete map can be generated.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ For example, it turns out that the ratio of the length of two succeeding bifurcation branches is a universal constant that is applicable to all systems that exhibit this behavior.
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.December 2009" style="white-space:nowrap;">[citation needed] Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite-dimensional.^ To apply Chaos Theory, a single measured variable x(n) = x(t0 + nt) with a starting time, t0, and a lead time, t, provides an n-dimensional space, or phase space, that represents the full multivariate state space of the system; up to 4 dimensions may be required to represent the phase space for a chaotic system.
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^ A system exhibiting nonlinear behavior may appear quite random over time, yet studies of chaotic regimes in phase-space reveal underlying patterns.
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^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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[31] .A theory of linear chaos is being developed in the functional analysis, a branch of mathematical analysis.^ "Based on chaos theory, Dr. Marcial Losada has developed a method to improve High Performance Teams and has established the "Losada Line" using non linear mathematics.
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^ Unfortunately, this would mean that chaos theory would be only a mathematical theory and not a physical one.
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^ Chaos theory is once and awhile at risk of being overtaxed by being associated with everything that can be even superficially related to the concept of chaos (9).
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History

.
Fractal fern created using the chaos game.
^ Fractals and modern chaos theory are also linked by the fact that many of the contemporary pace-setting discoveries in their fields were only possible using computers.
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.Natural forms (ferns, clouds, mountains, etc.^ Depicting natural things such as clouds and mountain ranges has been a challenge for computer graphics systems based on everyday geometry.
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) may be recreated through an Iterated function system (IFS).
The first discoverer of chaos was Henri Poincaré. .In the 1880s, while studying the three-body problem, he found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point.^ It was from this approach he was able to show that the three-body problem has complicated orbital dynamics, which we now call chaos.
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^ He gives three points of contrast between this approach to understanding and what he takes to be the standard approach to understanding in the sciences.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Rather than following all the trajectories of every orbit, he instead worked out a geometric approach to investigate the problem.
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[32][33] In 1898 Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature.[34] .In the system studied, "Hadamard's billiards," Hadamard was able to show that all trajectories are unstable in that all particle trajectories diverge exponentially from one another, with a positive Lyapunov exponent.^ For systems exhibiting SDIC, trajectories starting out in a highly localized region of state space will diverge on-average exponentially fast from one another.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ But in the long run, some of these trajectories could effectively diverge as if there was on-average exponential growth in uncertainties as characterized by global Lyapunov exponents.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Therefore, drawing an inference from the presence of positive global Lyapunov exponents to the existence of on-average exponentially diverging trajectories is invalid.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.Much of the earlier theory was developed almost entirely by mathematicians, under the name of ergodic theory.^ Indeed another great mathematician named Gaston Julia came up with a long memoir developing the theory in a similar way to Fatou.
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.Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff,[35] A. N. Kolmogorov,[36][37][38] M.L. Cartwright and J.E. Littlewood,[39] and Stephen Smale.^ Number theory and the study of non-linear differential equations also give examples of fractal sets.
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^ The dynamical systems of interest in chaos studies are nonlinear , such as the Lorenz model equations for convection in fluids: .
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Computers permitted one to experiment with the nonlinear differential equations that were unfeasible before.
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[40] .Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood.^ Except for Smale, who was perhaps the first pure mathematician to study nonlinear dynamics, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood.
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^ Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood.
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^ At the end of the 19th century, the French mathematician Henri Poincare tried to solve the differential equations for the three body problem.

[citation needed] .Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.^ Turbulence is the chaotic flow of a fluid.

^ Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.
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^ Their findings supported the theory that chaotic attractors cause fluid turbulence.
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.Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the logistic map.^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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^ Chaos theory is a way to analyze such systems.
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^ To apply Chaos Theory, a single measured variable x(n) = x(t0 + nt) with a starting time, t0, and a lead time, t, provides an n-dimensional space, or phase space, that represents the full multivariate state space of the system; up to 4 dimensions may be required to represent the phase space for a chaotic system.
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.What had been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories as a full component of the studied systems.^ Two more inches might greatly increase the number of parking violations; then even 1/2 inch more might throw the whole system into full blown chaos.
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^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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^ Chaos Theory: A science which deals with the complex harmonies and dis-harmonies exhibited by natural and social systems.
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.The main catalyst for the development of chaos theory was the electronic computer.^ The arrival of digital computers has accelerated the pace of development in the field of chaos.
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^ Fractals and modern chaos theory are also linked by the fact that many of the contemporary pace-setting discoveries in their fields were only possible using computers.
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^ "Based on chaos theory, Dr. Marcial Losada has developed a method to improve High Performance Teams and has established the "Losada Line" using non linear mathematics.
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.Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand.^ Unfortunately, this would mean that chaos theory would be only a mathematical theory and not a physical one.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Chaos theory would suggest that a parking regime with more than 6 or 8 such basins would be close to a bifurcation point at which 'illegal' parking would begin to increase qualitatively.
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^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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.Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems.^ Electronic computers made these repeated calculations practical.
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^ Complex systems are systems that contain so much motion (so many elements that move) that computers are required to calculate all the various possibilities.

^ These system brochures often contain stories of legendary or reclusive traders who discovered secrets of the markets and made millions.
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Turbulence in the tip vortex from an airplane wing. .Studies of the critical point beyond which a system creates turbulence was important for Chaos theory, analyzed for example by the Soviet physicist Lev Landau who developed the Landau-Hopf theory of turbulence.^ Chaos theory is a way to analyze such systems.
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^ For this reason, we can say that the transition from laminar flow to turbulence is an initiation of self-organization --it is the creation of order from chaos.
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^ Chaos theory would suggest that a parking regime with more than 6 or 8 such basins would be close to a bifurcation point at which 'illegal' parking would begin to increase qualitatively.
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David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory.
.An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961.[41] Lorenz was using a simple digital computer, a Royal McBee LGP-30, to run his weather simulation.^ In 1960 he was working on the dilemma of weather prediction.
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^ It is considered that Edward Lorenz a meteorologist was the first true experimenter in chaos.
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^ The first such attractor was identified by Edward Lorenz in his meteorology simulations in the early 1960s (Lorenz, 1963).
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.He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course.^ He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course.
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^ To save time, he started in the middle of the sequence, instead of the beginning.
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^ He started in the middle of the first number sequence.
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.He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.^ He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.
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^ He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course.
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^ Using real time disaster response data or a simulated exercise, test Koehler's and Comfort's hypothesis about how disaster wide networks emerge .
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.To his surprise the weather that the machine began to predict was completely different from the weather calculated before.^ To View Full Size Images Click Image 1 Image 2 To his surprise the weather that the machine began to predict was completely different to the weather calculated before.
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^ To his surprise the weather that the machine began to predict was completely different from the weather calculated before.
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^ The time it takes to complete the next order cannot be predicted because different processing rules may apply and/or the complexity of the order itself can not be predicted.
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.Lorenz tracked this down to the computer printout.^ Lorenz tracked this down to the computer printout.
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The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 was printed as 0.506. .This difference is tiny and the consensus at the time would have been that it should have had practically no effect.^ This difference is tiny and the consensus at the time would have been that it should have had practically no effect.
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^ There's usually no practical difference to the approach, so it's mostly just a matter of personal preference, but at least you now have the choice.
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^ The butterfly effect can be better understood by examining the two different time series in figure 4.
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.However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.^ However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.The term chaos as used in mathematics was coined by the applied mathematician James A. Yorke.
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^ The idea is that small variations in the initial conditions of a dynamical system produce large variations in the long term behavior of the system.
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^ However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.
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[42] .Lorenz's discovery, which gave its name to Lorenz attractors, proved that meteorology could not reasonably predict weather beyond a weekly period (at most).^ Yet there was little reason to doubt that precise predictability could, in principle, be achieved.
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^ From that idea Lorenz stated that it is impossible to predict the weather accurately.
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^ Even though he couldnt predict the weather, his discovery led Lorenz on to other aspects of what eventually came to be known as chaos theory.
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.The year before, Benoît Mandelbrot found recurring patterns at every scale in data on cotton prices.^ Mandelbrot has found order in places where others before him saw only chaos.
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^ Before turning to a more detailed discussion of these three states, does the logistic equation when applied to disaster response data fit this pattern of behavior ?
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[43] .Beforehand, he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion of noise-containing periods to error-free periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy.^ Number theory and the study of non-linear differential equations also give examples of fractal sets.
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[44] .Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur, e.g., in a stock's prices after bad news, thus challenging normal distribution theory in statistics, aka Bell Curve) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards).^ I'm not a scientist but I think the great solutions for our planet depend on a new way of thinking about changes and effects.
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^ Chaos theory thus adds understanding and support to the logic of many new management ideas.
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^ Both higher prices and higher taxes are required thus reducing competitive advantages and neglecting essential common social needs.
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[45][46] In 1967, he published "How long is the coast of Britain? .Statistical self-similarity and fractional dimension," showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an infinitesimally small measuring device.^ The self-similar structure is repeated on arbitrarily small scales .
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ In a similar study, Matsushita, Sano, Hayakawa, Honjo and Sawada measured the dimension of metallic zinc deposits to be 1.7, which agreed with the dimension of the computer-generated fractal.
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^ An important geometric implication of self-similarity is that there is no inherent size scale so that we can take as large a magnification of as small a region of the attractor as we want and a statistically similarly structure will be repeated (Hilborn 1994, p.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

[47] .Arguing that a ball of twine appears to be a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional.^ The important point is that at or near the boundary of chaos it appears that the ordered structure of the disaster response agency loosens, potentially making new behavior possible.
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.An object whose irregularity is constant over different scales ("self-similarity") is a fractal (for example, the Koch curve or "snowflake", which is infinitely long yet encloses a finite space and has fractal dimension equal to circa 1.2619, the Menger sponge and the Sierpiński gasket).^ Thus the Koch snowflake is an infinitely long curve that encloses a finite area.
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^ As can be observed, the fractal curve is self-similar.
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^ The Koch snowflake clearly illustrates the concept of fractal dimension.
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.In 1975 Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos theory.^ Chaos theory has also given us fractal geometry, the visual representation of chaos theory.
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^ Classical physics, even including the advanced ramifications of quantum theory, deals primarily with ordered systems, but the natural world exhibits a tendency toward disorder.
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^ Still since it is describable by deterministic equations, chaos theory supports a strictly deterministic philosophy of nature, although within subtle epistemic limits.

.Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.^ The idea here is that if a model is faithful in reproducing the behavior of the target system, refining the model will produce an even better fit with the target system's behavior.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ One possibility is to look for hypotheses about how such models are deployed when studying real physical systems.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Perhaps the fractal character of strange attractors is an artifact introduced through the various idealizations and approximations used to derive such chaotic models.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

Chaos was observed by a number of experimenters before it was recognized; e.g., in 1927 by van der Pol[48] and in 1958 by R.L. Ives.[49][50] However, as a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers (that is, vacuum tubes) and noticed, on Nov. 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.[51][52]
.In December 1977 the New York Academy of Sciences organized the first symposium on Chaos, attended by David Ruelle, Robert May, James A. Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw (a physicist, part of the Eudaemons group with J. Doyne Farmer and Norman Packard who tried to find a mathematical method to beat roulette, and then created with them the Dynamical Systems Collective in Santa Cruz, California), and the meteorologist Edward Lorenz.^ To begin, chaos is typically understood as a mathematical property of a dynamical system .
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ The name chaos was coined by Jim Yorke , an applied mathematician at the University of Maryland (Ruelle, 1991).
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^ I will return to the question of using mathematical models to represent real-world systems throughout this article.
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.The following year, Mitchell Feigenbaum published the noted article "Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic maps.^ Several years later May found an instance of chaos in the iterative map (logistic map) that was used to study population dynamics in biology.
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^ Immediately following a disaster organizations occupy different places on the logistic map.
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^ Kellert's (1993) focus on chaos models is suggestive of the semantic view of theories, and many texts and articles on chaos focus on models (e.g., logistic map, Henon map, Lorenz attractor).
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

[53] .Feigenbaum had applied fractal geometry to the study of natural forms such as coastlines.^ Most forms that occur in nature are fractals.
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^ Fractal geometry has even been applied to computer-aided music composition.
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^ Corollary to such work is the need to cultivate a mathematics and a geometry with which to study such dynamics.
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.Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different phenomena.^ Many of us have heard of “ Chaos Theory “.
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^ Application of chaos theory to family interaction .
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^ Chaos Theory is no different.
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.In 1979, Albert J. Libchaber, during a symposium organized in Aspen by Pierre Hohenberg, presented his experimental observation of the bifurcation cascade that leads to chaos and turbulence in convective Rayleigh–Benard systems.^ It is usually depicted as a fork in the time sequence of a system, in which a system can take two possible branches, one or both leading to chaos.
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^ For this reason, we can say that the transition from laminar flow to turbulence is an initiation of self-organization --it is the creation of order from chaos.
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^ The dynamical systems of interest in chaos studies are nonlinear , such as the Lorenz model equations for convection in fluids: .
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He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J. Feigenbaum "for his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems".[54]
.Then in 1986 the New York Academy of Sciences co-organized with the National Institute of Mental Health and the Office of Naval Research the first important conference on Chaos in biology and medicine.^ William Morrow and Co.: New York, 1990.
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^ The important point is that at or near the boundary of chaos it appears that the ordered structure of the disaster response agency loosens, potentially making new behavior possible.
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^ Chaos theory offers management science insight about where and when management control is reasonable or possible and at what scale of organization such control efforts are best directed.
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.There, Bernardo Huberman presented a mathematical model of the eye tracking disorder among schizophrenics.^ On the other hand, if a mathematical model is merely mimicking the behavior of a target system, there is no guarantee that the model has any genuine correspondence to the actual qualities of the target system.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Or is there really no one-to-one relationship between our mathematical models and target systems?
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

[55] .This led to a renewal of physiology in the 1980s through the application of chaos theory, for example in the study of pathological cardiac cycles.^ Limitations Of Chaos Theory For Disaster Management The application of chaos theory to disaster management has a number of significant limitations: The question of how helpful chaos theory is for disaster management needs further empirical investigation.
  • WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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^ Such research should seek to link existing sociological, psychological and other organizational studies of disasters to chaos theory.
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.In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters[56] describing for the first time self-organized criticality (SOC), considered to be one of the mechanisms by which complexity arises in nature.^ The capacity to develop a self-organizing response involves the whole system and varying social time from top to bottom.
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^ How various agencies link together, their social times, their ability to self-organize and mobilize resources are all directly related to how quickly the response emerges.
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^ This suggests, that attention to social time entrainment and self organizing processes--what is necessary within a particular context to bring about a rapidly developing overall response--is critical, rather than attention to quantitative objectives alone.
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.Alongside largely lab-based approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have centered around large-scale natural or social systems that are known (or suspected) to display scale-invariant behaviour.^ This property is known as scale invariance .
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^ The Emergence Of Geographically Distributed Large Scale Response Systems Up to this point the discussion has focused on how organizations and systems are disrupted by changes in their environment.
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^ Chaos Theory: A science which deals with the complex harmonies and dis-harmonies exhibited by natural and social systems.
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.Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including: earthquakes (which, long before SOC was discovered, were known as a source of scale-invariant behaviour such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the Omori law[57] describing the frequency of aftershocks); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires; landslides; epidemics; and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould).^ This property is known as scale invariance .
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^ The surfaces of certain cheeses and the random distribution of the stars in the sky display the property known as statistical self-similarity.
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^ It turns out that in mathematical theory the change from order and predictability into unpredictability or chaos for dynamic systems is governed by a single law, and that the 'route' between the two conditions is a universal one.
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.Worryingly, given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of wars.^ Of further interest is that some of Stanley's models suggest the unexpected result of an eventual increase in AIDS within groups that were previously consider "low-risk".
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^ This suggests then, as with variation beyond the control limits in quality measurement systems, that variation should not be considered a problem in management systems but rather an opportunity to learn why the variation occurred.
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^ Consider the propagator, J ( x ( t )), a function that evolves trajectories x ( t ) in time (an example of a propagator is given in the Appendix ).
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.These "applied" investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.^ The model developed for this article is one of a two-adversary system.

^ Classical physics, even including the advanced ramifications of quantum theory, deals primarily with ordered systems, but the natural world exhibits a tendency toward disorder.
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^ In physics, though, chaotic systems are ‘classical’ in scale and thus subsumable in principle under classical mechanics with its deterministic laws of motion.

.The same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public.^ Yet, what is it that makes chaos theory so fascinating?
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^ Chaos theory isn't new to astronomers.
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^ Computer-generated images applying Chaos Theory principles.
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.At first the domain of work of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of nonlinear systems analysis.^ A nonlinear system is a system that is not in the first degree.
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^ After all, it also seems to be the case that realism for chaos models has more to do with processes—namely stretching and folding mechanisms at work in target systems.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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.Alluding to Thomas Kuhn's concept of a paradigm shift exposed in The Structure of Scientific Revolutions (1962), many "chaologists" (as some self-nominated themselves) claimed that this new theory was an example of such a shift, a thesis upheld by J. Gleick.^ Chaos theory has many new concepts with which to grasp its ideas.
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^ In 1971 Ruelle and Takens came up with a new theory, based on the abstract concept of a strange attractor, for the onset of turbulence in fluids.
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^ There is no emphasis on the precise structure of scientific theories in Kuhn's picture of science.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.The availability of cheaper, more powerful computers broadens the applicability of chaos theory.^ The availability of cheaper, more powerful computers broadens the applicability of chaos theory.
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^ Add More Products To chaos theory .
  • Amazon.com: chaos theory 19 January 2010 9:53 UTC www.amazon.com [Source type: General]

^ Chaos theory is great as a way of looking at events which happen in the world differently from the more traditional strictly deterministic view which has dominated science from Newtonian times.
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.Currently, chaos theory continues to be a very active area of research, involving many different disciplines (mathematics, topology, physics, population biology, biology, meteorology, astrophysics, information theory, etc.^ Chaos theory and nursing systems research.
  • Bibliography of Chaos Theory 19 January 2010 9:53 UTC www.southernct.edu [Source type: Academic]

^ Currently, chaos theory continues to be a very active area of research.
  • What is Chaos Theory? - Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]

^ Many of us have heard of “ Chaos Theory “.
  • Can You Tame Chaos Theory Into Believable Emergent Behaviors? —  AiGameDev.com 19 January 2010 9:53 UTC aigamedev.com [Source type: FILTERED WITH BAYES]

).
Chaos theory is employed in everyday life in some microwave ovens, to aid rapid and (most notably) evenly-spread defrosting using microwave energy; this function is known as Chaos Defrost and was first developed by Panasonic in 2001.[58]

Distinguishing random from chaotic data

.It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.'^ Time series data may be nonlinear but not chaotic.
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^ [Review of the article Can the analytic techniques of nonlinear dynamics distinguish periodic, random and chaotic signals?
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^ Obtaining and processing time series data to track and steer disaster response processes is difficult : Inputting data can be time consuming, require special skills, and take personnel away from providing immediately needed services, particularly at the beginning of the response.
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.There will always be some form of corrupting noise, even if it is present as round-off or truncation error.^ With this all-purpose flour recipe, there shouldn’t be much “bounceback” (when the round or oval pizza dough contracts back) but there may be some.
  • Chaos Theory | Pizza Goon Pizza Blog 19 January 2010 9:53 UTC pizzagoon.com [Source type: General]

^ But, pi cannot be calculated exactly; it must always be rounded off and is therefore nonlinear.
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^ So, in a nod of deference to the mysteries of mathematical chaos that should be in play, we add some parametric noise that don’t even attempt to model the above criteria.
  • Can You Tame Chaos Theory Into Believable Emergent Behaviors? —  AiGameDev.com 19 January 2010 9:53 UTC aigamedev.com [Source type: FILTERED WITH BAYES]

.Thus any real time series, even if mostly deterministic, will contain some randomness.^ Chaos thus looks like random behavior but is really unstable behavior over time that stays within clear boundaries.
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^ But even if some of the changes are marginal, this is still the most entertaining, most well-rounded game in the series yet.
  • GameFly: Tom Clancy's Splinter Cell Chaos Theory for Xbox Review and Ratings by IGN, GameSpot, and GameSpy 19 January 2010 9:53 UTC www.gamefly.com [Source type: General]

^ In such cases, after performing some tests on the data set, the modeler sets to work constructing a mathematical model that reproduces the time series as its output.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

[59]
.All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point.^ He could never start the chronographic loopback engine trials at exactly the same point in time because he could not freeze time.
  • Darker Matter - Chaos Theory by Rick Novy 19 January 2010 9:53 UTC www.darkermatter.com [Source type: Original source]

^ With a starting number of 2, the final result can be entirely different from the same system with a starting value of 2.000001.
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^ The logic of this point is that, in the future, administration of a given firm must be open to policies which consider the stability of the whole system.
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[59][60] Thus, given a time series to test for determinism, one can:
  1. pick a test state;
  2. search the time series for a similar or 'nearby' state; and
  3. compare their respective time evolutions.
.Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby state.^ We see that a small difference, a small error, or a small change can have very novel, unexpected and even explosive effects over time.
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^ When we see two branches this just means that the long term behavior of the system is now alternating between two different states, a lower one and an upper one.
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^ Because there is a fundamental difference between Time Spiral timeshifted cards and Planar Chaos timeshifted cards.
  • Chaos Theory : Daily MTG : Magic: The Gathering 19 January 2010 9:53 UTC www.wizards.com [Source type: Original source]

.A deterministic system will have an error that either remains small (stable, regular solution) or increases exponentially with time (chaos).^ It is usually depicted as a fork in the time sequence of a system, in which a system can take two possible branches, one or both leading to chaos.
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^ It gives us phenomena, models, and metaphors such as deterministic chaos, “strange attractors,” turbulence, stretching time and folding space, and nonlinear phenomena.
  • Chaos Theory & Complexes « Iona Miller's Weblog 19 January 2010 9:53 UTC ionamiller.wordpress.com [Source type: FILTERED WITH BAYES]

^ We see that a small difference, a small error, or a small change can have very novel, unexpected and even explosive effects over time.
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A stochastic system will have a randomly distributed error.[61]
.Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state (i.e., correlation dimension, Lyapunov exponents, etc.^ So analysts need to be careful not to call all "messy" looking time series data as chaotic.
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^ You can find almost all the nike series there, in huge collection and varies colorways.
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^ This has implications for defining chaos because exponential growth parametrized by global Lyapunov exponents turn out to not be an appropriate measure.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

). .To define the state of a system one typically relies on phase space embedding methods.^ Any state of the system at a moment in time is represented as a point in phase space.
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^ A phase space diagram is a history of the changing variables of the system.
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^ Because of SDIC, nonlinear chaotic systems whose initial states can be located only within a small neighborhood ε of state space will have future states that can be located only within a much larger patch δ.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

[62] .Typically one chooses an embedding dimension, and investigates the propagation of the error between two nearby states.^ Like the flow in the rotating dishpan, the system had at least two states that it could flip between for no obvious reason.
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^ When we see two branches this just means that the long term behavior of the system is now alternating between two different states, a lower one and an upper one.
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^ In reality, the vortex oscillates between two states or phases of motion as it spins.
  • DocWeather : The Weather Eye > Weather and Chaos Theory: the Standard Map 19 January 2010 9:53 UTC docweather.com [Source type: FILTERED WITH BAYES]

If the error looks random, one increases the dimension. If you can increase the dimension to obtain a deterministic looking error, then you are done. .Though it may sound simple it is not really.^ As such, this defining characteristic could be applied to both mathematical models and real-world systems, though the identification of such mechanisms in target systems may be rather tricky.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.One complication is that as the dimension increases the search for a nearby state requires a lot more computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) to find a suitably close candidate.^ If the environment becomes even more disordered requiring the commitment of even more resources or their exhaustion the organization is forced to occupy any one of four structural states, then eight, until chaos sets in.
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^ Before turning to a more detailed discussion of these three states, does the logistic equation when applied to disaster response data fit this pattern of behavior ?
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^ LEARN MORE Data Center Services Plan, Launch and Manage Your Data Centers More Efficiently Dynamic Infrastructure requires more innovative, nimble and flexible data centers.
  • When Legal Strikes—Chaos Theory Meets DRM - Security from eWeek 19 January 2010 9:53 UTC www.eweek.com [Source type: General]

.If the embedding dimension (number of measures per state) is chosen too small (less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the dimension too large – the method will work.^ There was one small problem.
  • Chaos Theory : Daily MTG : Magic: The Gathering 19 January 2010 9:53 UTC www.wizards.com [Source type: Original source]

^ Both the deterministic character of the kinetic equations whereby the set of possible states and their respective stability can be calculated, and the random fluctuations 'choosing' between or among the states around the bifurcation points are inextricably connected.

^ Even if a small refinement to the model is made “in the right direction”, there is no guarantee that the nonlinear model will monotonically improve in capturing the target system's behavior.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

.When a non-linear deterministic system is attended by external fluctuations, its trajectories present serious and permanent distortions.^ For many systems, the addition of the nth doubling will cause it to transform into non-linearity.
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^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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^ Chaos research studies the transitions between linear and non-linear states of such dynamical systems.
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.Furthermore, the noise is amplified due to the inherent non-linearity and reveals totally new dynamical properties.^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
  • Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]

^ Chaos research studies the transitions between linear and non-linear states of such dynamical systems.
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^ According to respected authorities, stock markets are non-linear, dynamic systems.
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Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate the deterministic part risk failure. .Things become worse when the deterministic component is a non-linear feedback system.^ For many systems, the addition of the nth doubling will cause it to transform into non-linearity.
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^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
  • Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]

^ Chaos research studies the transitions between linear and non-linear states of such dynamical systems.
  • TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]

[63] .In presence of interactions between nonlinear deterministic components and noise, the resulting nonlinear series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture.^ It is then hoped that patterns of neural, cognitive and human activity can be explained as the results of nonlinear dynamical processes involving causal interactions and constraints at multiple levels (e.g., neurons, brains, bodies, physical environments).
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

^ Nonlinearity refers to behavior in which the relationships between variables in a system are dynamic and disproportionate.
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^ Stephen Kellert defines chaos theory as “the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems” (1993, p.
  • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

[64]

Cultural references

.Chaos theory has been mentioned in numerous novels and movies, such as Jurassic Park.^ Chaos theory would suggest that a parking regime with more than 6 or 8 such basins would be close to a bifurcation point at which 'illegal' parking would begin to increase qualitatively.
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^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
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^ Such research should seek to link existing sociological, psychological and other organizational studies of disasters to chaos theory.
  • WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

See also

References

  1. ^ Stephen H. Kellert, In the Wake of Chaos: Unpredictable Order in Dynamical Systems, University of Chicago Press, 1993, p 32, ISBN 0226429768.
  2. ^ Kellert, p. 56.
  3. ^ Kellert, p. 62.
  4. ^ a b Raymond Sneyers (1997) "Climate Chaotic Instability: Statistical Determination and Theoretical Background", Environmetrics, vol. 8, no. 5, pages 517–532.
  5. ^ Kyrtsou, C. and W. Labys, (2006). Evidence for chaotic dependence between US inflation and commodity prices, Journal of Macroeconomics, 28(1), pp. 256–266.
  6. ^ Kyrtsou, C. and W. Labys, (2007). Detecting positive feedback in multivariate time series: the case of metal prices and US inflation, Physica A, 377(1), pp. 227–229.
  7. ^ Kyrtsou, C., and Vorlow, C., (2005). Complex dynamics in macroeconomics: A novel approach, in New Trends in Macroeconomics, Diebolt, C., and Kyrtsou, C., (eds.), Springer Verlag.
  8. ^ Applying Chaos Theory to Embedded Applications
  9. ^ Hristu-Varsakelis, D., and Kyrtsou, C., (2008): Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns, Discrete Dynamics in Nature and Society, Volume 2008, Article ID 138547, 7 pages, doi:10.1155/2008/138547.
  10. ^ Kyrtsou, C. and M. Terraza, (2003). Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass equation with heteroskedastic errors to the Paris Stock Exchange returns series, Computational Economics, 21, 257–276.
  11. ^ Metaculture.net, metalinks: Applied Chaos, 2007.
  12. ^ Apostolos Serletis and Periklis Gogas,Purchasing Power Parity Nonlinearity and Chaos, in: Applied Financial Economics, 10, 615–622, 2000.
  13. ^ Apostolos Serletis and Periklis Gogas The North American Gas Markets are ChaoticPDF (918 KB), in: The Energy Journal, 20, 83–103, 1999.
  14. ^ Apostolos Serletis and Periklis Gogas, Chaos in East European Black Market Exchange Rates, in: Research in Economics, 51, 359–385, 1997.
  15. ^ Comdig.org, Complexity Digest 199.06
  16. ^ Michael Berry, "Quantum Chaology," pp 104-5 of Quantum: a guide for the perplexed by Jim Al-Khalili (Weidenfeld and Nicolson 2003), http://www.physics.bristol.ac.uk/people/berry_mv/the_papers/Berry358.pdf.
  17. ^ A. E. Motter, Relativistic chaos is coordinate invariant, in: Phys. Rev. Lett. 91, 231101 (2003).
  18. ^ Definition of chaos at Wiktionary.
  19. ^ Hasselblatt, Boris; Anatole Katok (2003). A First Course in Dynamics: With a Panorama of Recent Developments. .Cambridge University Press.^ Smith, P. (1998), Explaining Chaos , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The Large, the Small and the Human Mind , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Ott, E. (2002), Chaos in Dynamical Systems , Cambridge: Cambridge University Press, 2 nd edition.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ISBN 0521587506.
     
  20. ^ Saber N. Elaydi, Discrete Chaos, Chapman & Hall/CRC, 1999, page 117, ISBN 1584880023.
  21. ^ William F. Basener, Topology and its applications, Wiley, 2006, page 42, ISBN 0471687553,
  22. ^ Michel Vellekoop; Raoul Berglund, "On Intervals, Transitivity = Chaos," The American Mathematical Monthly, Vol. 101, No. 4. (April, 1994), pp. 353–355 [1]
  23. ^ Alfredo Medio and Marji Lines, Nonlinear Dynamics: A Primer, Cambridge University Press, 2001, page 165, ISBN 0521558743.
  24. ^ Robert G. Watts, Global Warming and the Future of the Earth, Morgan & Claypool, 2007, page 17.
  25. ^ Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems, 2nd ed. Westview Press. ISBN 0-8133-4085-3. 
  26. ^ Alligood, K. T., Sauer, T., and Yorke, J.A. (1997). .Chaos: an introduction to dynamical systems.^ After all, chaos is typically understood to be a property of the dynamics of such systems, and dynamics is usually determined by the processes at work and their interactions.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The question of defining chaos is basically the question what makes a dynamical system like (1) chaotic rather than nonchaotic.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
    • Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]

    Springer-Verlag New York, LLC. ISBN 0-387-94677-2.
     
  27. ^ Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." American Mathematical Monthly 82, 985–92, 1975.[2]
  28. ^ Sprott, J.C. (1997). "Simplest dissipative chaotic flow". Physics Letters A 228: 271. doi:10.1016/S0375-9601(97)00088-1. 
  29. ^ Fu, Z.; Heidel, J. (1997). "Non-chaotic behaviour in three-dimensional quadratic systems". Nonlinearity 10: 1289. doi:10.1088/0951-7715/10/5/014. 
  30. ^ Heidel, J.; Fu, Z. (1999). "Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case". Nonlinearity 12: 617. doi:10.1088/0951-7715/12/3/012. 
  31. ^ Bonet, J.; Martínez-Giménez, F.; Peris, A. (2001). "A Banach space which admits no chaotic operator". Bulletin of the London Mathematical Society 33: 196-198. doi:10.1112/blms/33.2.196. 
  32. ^ Jules Henri Poincaré (1890) "Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt," Acta Mathematica, vol. 13, pages 1–270.
  33. ^ Florin Diacu and Philip Holmes (1996) Celestial Encounters: The Origins of Chaos and Stability, Princeton University Press.
  34. ^ Hadamard, Jacques (1898). "Les surfaces à courbures opposées et leurs lignes géodesiques". Journal de Mathématiques Pures et Appliquées 4: pp. 27–73. 
  35. ^ George D. Birkhoff, Dynamical Systems, vol. 9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)
  36. ^ Kolmogorov, Andrey Nikolaevich (1941). "Local structure of turbulence in an incompressible fluid for very large Reynolds numbers". Doklady Akademii Nauk SSSR 30 (4): 301–305.  Reprinted in: Proceedings of the Royal Society of London: Mathematical and Physical Sciences (Series A), vol. 434, pages 9–13 (1991).
  37. ^ Kolmogorov, A. N. (1941). "On degeneration of isotropic turbulence in an incompressible viscous liquid". Doklady Akademii Nauk SSSR 31 (6): 538–540.  Reprinted in: Proceedings of the Royal Society of London: Mathematical and Physical Sciences (Series A), vol. 434, pages 15–17 (1991).
  38. ^ Kolmogorov, A. N. (1954). "Preservation of conditionally periodic movements with small change in the Hamiltonian function". Doklady Akademii Nauk SSSR 98: 527–530.  See also Kolmogorov–Arnold–Moser theorem
  39. ^ Mary L. Cartwright and John E. Littlewood (1945) "On non-linear differential equations of the second order, I: The equation y" + k(1−y2)y' + y = bλkcos(λt + a), k large," Journal of the London Mathematical Society, vol. 20, pages 180–189. See also: Van der Pol oscillator
  40. ^ Stephen Smale (January 1960) "Morse inequalities for a dynamical system," Bulletin of the American Mathematical Society, vol. 66, pages 43–49.
  41. ^ Edward N. Lorenz, "Deterministic non-periodic flow," Journal of the Atmospheric Sciences, vol. 20, pages 130–141 (1963).
  42. ^ Gleick, James (1987). Chaos: Making a New Science. London: Cardinal. pp. 17. 
  43. ^ Mandelbrot, Benoît (1963). "The variation of certain speculative prices". Journal of Business 36: pp. 394–419. 
  44. ^ J.M. Berger and B. Mandelbrot (July 1963) "A new model for error clustering in telephone circuits," I.B.M. Journal of Research and Development, vol 7, pages 224–236.
  45. ^ B. Mandelbrot, The Fractal Geometry of Nature (N.Y., N.Y.: Freeman, 1977), page 248.
  46. ^ See also: Benoît B. Mandelbrot and Richard L. Hudson, The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward (N.Y., N.Y.: Basic Books, 2004), page 201.
  47. ^ Benoît Mandelbrot (5 May 1967) "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension," Science, Vol. 156, No. 3775, pages 636–638.
  48. ^ B. van der Pol and J. van der Mark (1927) "Frequency demultiplication," Nature, vol. 120, pages 363–364. See also: Van der Pol oscillator
  49. ^ R.L. Ives (10 October 1958) "Neon oscillator rings," Electronics, vol. 31, pages 108–115.
  50. ^ See p. 83 of Lee W. Casperson, "Gas laser instabilities and their interpretation," pages 83–98 in: N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds., Instabilities and Chaos in Quantum Optics II: Proceedings of the NATO Advanced Study Institute, Il Ciocco, Italy, June 28–July 7, 1987 (N.Y., N.Y.: Springer Verlag, 1988).
  51. ^ Ralph H. Abraham and Yoshisuke Ueda, eds., The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory (Singapore: World Scientific Publishing Co., 2001). See Chapters 3 and 4.
  52. ^ Sprott, J. Chaos and time-series analysis. Oxford. University Press, Oxford, UK, & New York, USA. 2003
  53. ^ Mitchell Feigenbaum (July 1978) "Quantitative universality for a class of nonlinear transformations," Journal of Statistical Physics, vol. 19, no. 1, pages 25–52.
  54. ^ "The Wolf Prize in Physics in 1986.". http://www.wolffund.org.il/cat.asp?id=25&cat_title=PHYSICS. 
  55. ^ Bernardo Huberman, "A Model for Dysfunctions in Smooth Pursuit Eye Movement" Annals of the New York Academy of Sciences, Vol. 504 Page 260 July 1987, Perspectives in Biological Dynamics and Theoretical Medicine
  56. ^ Per Bak, Chao Tang, and Kurt Wiesenfeld, "Self-organized criticality: An explanation of the 1/f noise," Physical Review Letters, vol. 59, no. 4, pages 381–384 (27 July 1987). However, the conclusions of this article have been subject to dispute. See: http://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html . See especially: Lasse Laurson, Mikko J. Alava, and Stefano Zapperi, "Letter: Power spectra of self-organized critical sand piles," Journal of Statistical Mechanics: Theory and Experiment, 0511, L001 (15 September 2005).
  57. ^ F. Omori (1894) "On the aftershocks of earthquakes," Journal of the College of Science, Imperial University of Tokyo, vol. 7, pages 111–200.
  58. ^ [3]The Independent, Fri 29 August 2003
  59. ^ a b Provenzale A. et al.: "Distinguishing between low-dimensional dynamics and randomness in measured time-series", in: Physica D, 58:31–49, 1992
  60. ^ Sugihara G. and May R.: "Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series", in: Nature, 344:734–41, 1990
  61. ^ Casdagli, Martin. "Chaos and Deterministic versus Stochastic Non-linear Modelling", in: Journal Royal Statistics Society: Series B, 54, nr. 2 (1991), 303-28
  62. ^ Broomhead D. S. and King G. P.: "Extracting Qualitative Dynamics from Experimental Data", in: Physica 20D, 217–36, 1986
  63. ^ Kyrtsou, C., (2008). Re-examining the sources of heteroskedasticity: the paradigm of noisy chaotic models, Physica A, 387, pp. 6785–6789.
  64. ^ Kyrtsou, C., (2005). Evidence for neglected linearity in noisy chaotic models, International Journal of Bifurcation and Chaos, 15(10), pp. 3391–3394.

Scientific literature

Articles

.
  • A.N. Sharkovskii, "Co-existence of cycles of a continuous mapping of the line into itself", Ukrainian Math.^ Examining the logistic map, we see that a single line curves from the origin up to a point where it divides or bifurcates into two lines.
    • WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    J., 16:61–71 (1964)
  • Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." American Mathematical Monthly 82, 985–92, 1975.
  • Kolyada, S. F. "Li-Yorke sensitivity and other concepts of chaos", Ukrainian Math. J. 56 (2004), 1242–1257.

Textbooks

  • Alligood, K. T., Sauer, T., and Yorke, J.A. (1997). .Chaos: an introduction to dynamical systems.^ After all, chaos is typically understood to be a property of the dynamics of such systems, and dynamics is usually determined by the processes at work and their interactions.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The question of defining chaos is basically the question what makes a dynamical system like (1) chaotic rather than nonchaotic.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Chaos theory is the mathematics of studying such non-linear, dynamic systems.
    • Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]

    .Springer-Verlag New York, LLC. ISBN 0-387-94677-2. 
  • Baker, G. L. (1996).^ Chaos and Order in Nature , New York: Springer, pp.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ New York: Springer.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ New York:Springer-Verlag, 1994.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    Chaos, Scattering and Statistical Mechanics. .Cambridge University Press.^ Smith, P. (1998), Explaining Chaos , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The Large, the Small and the Human Mind , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Ott, E. (2002), Chaos in Dynamical Systems , Cambridge: Cambridge University Press, 2 nd edition.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ISBN 0-521-39511-9. 
  • Badii, R.; Politi A. (1997). "Complexity: hierarchical structures and scaling in physics". .Cambridge University Press.^ Smith, P. (1998), Explaining Chaos , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The Large, the Small and the Human Mind , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Ott, E. (2002), Chaos in Dynamical Systems , Cambridge: Cambridge University Press, 2 nd edition.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ISBN 0521663857. http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521663857.
     
  • Devaney, Robert L. (2003). .An Introduction to Chaotic Dynamical Systems, 2nd ed,.^ More exactly, chaotic dynamical systems are characterized by "sensitive dependence on initial conditions".
    • WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ Thats what will be discussed more, is how the simple dynamical systems can react or behave in a very strange and chaotic way.
    • Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]

    ^ Towards an Interpretation of Dynamic Neural Activity in Terms of Chaotic Dynamical Systems”, Behavioral and Brain Sciences , 24: 793–847.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    Westview Press. ISBN 0-8133-4085-3.
     
  • Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. .Cambridge University Press.^ Smith, P. (1998), Explaining Chaos , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The Large, the Small and the Human Mind , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Ott, E. (2002), Chaos in Dynamical Systems , Cambridge: Cambridge University Press, 2 nd edition.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ISBN 0-521-47685-2.
     
  • Guckenheimer, J., and Holmes P. (1983). .Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.^ Kiel, L. D. "Nonlinear Dynamical Analysis: Assessing Systems Concepts in a Government Agency."
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ There is much more that will be learned over the next several decades about complex systems and nonlinear dynamics as scholars and managers understand more about this new vision of reality.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ One tenet of nonlinear dynamics is that complex systems defy simple formulation and thus may preclude the development of precise mathematical algorithms.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    .Springer-Verlag New York, LLC. ISBN 0-387-90819-6. 
  • Gutzwiller, Martin (1990).^ William Morrow and Co.: New York, 1990.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ Chaos and Order in Nature , New York: Springer, pp.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ New York: Springer.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    .Chaos in Classical and Quantum Mechanics.^ Chaos, Determinism and Quantum Mechanics .
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Moreover, the possible constraints of nonlinear classical mechanics systems on the amplification of quantum effects must be considered on a case-by-case basis.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ SD arguments purport to demonstrate that chaos in classical systems can amplify quantum fluctuations due to sensitivity to the smallest changes in initial conditions.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    .Springer-Verlag New York, LLC. ISBN 0-387-97173-4. 
  • Hoover, William Graham (1999,2001).^ William Morrow and Co.: New York, 1990.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ Chaos and Order in Nature , New York: Springer, pp.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ New York: Springer.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    Time Reversibility, Computer Simulation, and Chaos. World Scientific. ISBN 981-02-4073-2. 
  • Kiel, L. Douglas; Elliott, Euel W. (1997). .Chaos Theory in the Social Sciences.^ For the majority, chaos theory already belongs to the greatest achievements in the natural sciences in the twentieth century (9).
    • Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]

    ^ Chaos Theory, The Rules Of Organizational Morphogenesis, And Social Time Disaster response organizations and response systems are dynamic systems.
    • WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ Our understanding of how chaos theory, nonlinearity and instability and the sciences of complexity can help us better manage organizations is in its initial stages.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    Perseus Publishing. ISBN 0-472-08472-0.
     
  • Moon, Francis (1990). Chaotic and Fractal Dynamics. .Springer-Verlag New York, LLC. ISBN 0-471-54571-6. 
  • Ott, Edward (2002).^ New York: Springer.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ New York:Springer-Verlag, 1994.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ Chaos and Order in Nature , New York: Springer, pp.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    .Chaos in Dynamical Systems.^ Robinson, C. (1995), Dynamical Systems: Stability, Symbol Dynamics and Chaos , London: CRC Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ After all, chaos is typically understood to be a property of the dynamics of such systems, and dynamics is usually determined by the processes at work and their interactions.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The question of defining chaos is basically the question what makes a dynamical system like (1) chaotic rather than nonchaotic.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    .Cambridge University Press New, York.^ Smith, P. (1998), Explaining Chaos , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The Large, the Small and the Human Mind , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Ott, E. (2002), Chaos in Dynamical Systems , Cambridge: Cambridge University Press, 2 nd edition.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ISBN 0-521-01084-5. 
  • Strogatz, Steven (2000). .Nonlinear Dynamics and Chaos.^ Chaos, then, appears to be unstable aperiodic behavior in nonlinear dynamical systems.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Chaos, Variation, Learning, and Disaster Response Chaos theory teaches us of the value of variation as a means for learning has obvious relevance to the management lens of nonlinear dynamics.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ Hilborn, R. C. (1994), Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers , Oxford: Oxford University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    Perseus Publishing. ISBN 0-7382-0453-6.
     
  • Sprott, Julien Clinton (2003). .Chaos and Time-Series Analysis.^ Chaos Over Time A third type of nonlinear time series data that can be expected in the world of management is chaotic behavior.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ To examine the "order within chaos" in organizational data we can examine the "attractor" of time series data.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ The use of the term chaos here refers to total chaos when a system is completely disordered, and not to the chaos of time-series behavior used throughout this book.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    .Oxford University Press.^ Hilborn, R. C. (1994), Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers , Oxford: Oxford University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Kane, R. (1996), The Significance of Free Will , Oxford: Oxford University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Penrose, R. (1994) Shadows of the Mind , Oxford: Oxford University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ISBN 0-19-850840-9.
     
  • Tél, Tamás; Gruiz, Márton (2006). .Chaotic dynamics: An introduction based on classical mechanics.^ But it is neither clear how this would work in the case of nonlinear systems in classical mechanics, nor how this would work for chaotic models in biology, economics and other disciplines.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Furthermore, the relationship between the state spaces of chaotic models and the spaces of idealized physical systems is quite delicate, which seems to be a dissimilarity between classical mechanics and “chaos theory”.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    .Cambridge University Press.^ Smith, P. (1998), Explaining Chaos , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ The Large, the Small and the Human Mind , Cambridge: Cambridge University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Ott, E. (2002), Chaos in Dynamical Systems , Cambridge: Cambridge University Press, 2 nd edition.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ISBN 0-521-83912-2.
     
  • Tufillaro, Abbott, Reilly (1992). .An experimental approach to nonlinear dynamics and chaos.^ Chaos, then, appears to be unstable aperiodic behavior in nonlinear dynamical systems.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Chaos, Variation, Learning, and Disaster Response Chaos theory teaches us of the value of variation as a means for learning has obvious relevance to the management lens of nonlinear dynamics.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ Chaos and nonlinear dynamics are not only rich areas for scientific investigation, but also raise a number of interesting philosophical questions.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    Addison-Wesley New York. ISBN 0-201-55441-0.
     
  • Zaslavsky, George M. (2005). .Hamiltonian Chaos and Fractional Dynamics.^ If, in the case of Hamiltonian chaos, the dynamics is confined to an energy surface (by the action of a force like gravity), this surface could be spatially unbounded.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    .Oxford University Press.^ Hilborn, R. C. (1994), Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers , Oxford: Oxford University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Kane, R. (1996), The Significance of Free Will , Oxford: Oxford University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Penrose, R. (1994) Shadows of the Mind , Oxford: Oxford University Press.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ISBN 0-198-52604-0.
     

Semitechnical and popular works

  • Ralph H. Abraham and Yoshisuke Ueda (Ed.), .The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory, World Scientific Publishing Company, 2001, 232 pp.
  • Michael Barnsley, Fractals Everywhere, Academic Press 1988, 394 pp.
  • Richard J Bird, Chaos and Life: Complexity and Order in Evolution and Thought, Columbia University Press 2003, 352 pp.
  • John Briggs and David Peat, Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness, Harper Perennial 1990, 224 pp.
  • John Briggs and David Peat, Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change, Harper Perennial 2000, 224 pp.
  • Lawrence A. Cunningham, From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis, George Washington Law Review, Vol.^ It turns out that in mathematical theory the change from order and predictability into unpredictability or chaos for dynamic systems is governed by a single law, and that the 'route' between the two conditions is a universal one.
    • WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ PSA 1990, Volume 2, East Lansing: Philosophy of Science Association, pp.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Given our discussion of how unpredictable a disaster response is, chaos theory may be a useful way to understand this complex phenomena.
    • WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    .62, 1994, 546 pp.
  • Leon Glass and Michael C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press 1988, 272 pp.
  • James Gleick, Chaos: Making a New Science, New York: Penguin, 1988. 368 pp.
  • John Gribbin, Deep Simplicity,
  • L Douglas Kiel, Euel W Elliott (ed.^ New York: Simon and Schuster, 1988.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ New York:Springer-Verlag, 1994.
    • CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]

    ^ Oxford: Oxford University Press, pp.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ), .Chaos Theory in the Social Sciences: Foundations and Applications, University of Michigan Press, 1997, 360 pp.
  • Arvind Kumar, Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature , National Book Trust, 2003.
  • Hans Lauwerier, Fractals, Princeton University Press, 1991.
  • Edward Lorenz, The Essence of Chaos, University of Washington Press, 1996.
  • Chapter 5 of Alan Marshall (2002) The Unity of nature, Imperial College Press: London
  • Heinz-Otto Peitgen and Dietmar Saupe (Eds.^ It is considered that Edward Lorenz a meteorologist was the first true experimenter in chaos.
    • Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]

    ^ King, C. C. (1995), “Fractal Neurodynamics and Quantum Chaos: Resolving the Mind-Brain Paradox Through Novel Biophysics”, in E. MacCormack, E. and M.I. Stamenov, M.I. (eds.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ^ Questions about Realism and Explanation 5.1 Realism and Chaos 5.2 The Nature of Chaos Explanations 5.2.1 Explanation, Faithful Models and Chaos 5.2.2 Chaos and Understanding 5.2.3 Nothing New under the Sun 5.3 Taking Stock 6.
    • Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]

    ), The Science of Fractal Images, Springer 1988, 312 pp.
  • Clifford A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr 1991.
  • Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984.
  • Heinz-Otto Peitgen and P. H. Richter, The Beauty of Fractals : Images of Complex Dynamical Systems, Springer 1986, 211 pp.
  • David Ruelle, Chance and Chaos, Princeton University Press 1993.
  • Ivars Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993.
  • David Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989.
  • Peter Smith, Explaining Chaos, Cambridge University Press, 1998.
  • Ian Stewart, Does God Play Dice?: The Mathematics of Chaos , Blackwell Publishers, 1990.
  • Steven Strogatz, Sync: The emerging science of spontaneous order, Hyperion, 2003.
  • Yoshisuke Ueda, The Road To Chaos, Aerial Pr, 1993.
  • M. Mitchell Waldrop, Complexity : The Emerging Science at the Edge of Order and Chaos, Simon & Schuster, 1992.

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