From Wikipedia, the free encyclopedia
.^ Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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]
.^ One of the first characteristics applicable for warfare is extreme sensitivity to initial conditions, also known as the "butterfly effect."
.^ This makes the longterm evolution of the system impossible, in principle, to predict. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ This is unlikely to work, and impossible to work in the long term. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ Only a small change in the initial conditions can drastically change the longterm behavior of a system. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^{[1]} .^ What do these teams mean though? Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ These means involve the essential questions that drive recognition of both the problems and prospects for change in work processes. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]} .^ 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. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ These, and other factors mentioned by Koehler and Comfort below, may condition how quickly a disaster wide response system emerges. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^{[3]} .^ 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. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
.^ 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 realworld 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. CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^{[4]} .^ 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]
Applications
.^ 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. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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]
^{[11]}
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]
.^ The weather is another system that changes dramatically over time. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ 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]}
.^ Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ "How does one start using chaos theory to achieve a desired outcome? Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ RE: Using the chaos theory . Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
.^ Applied chaos theory: A paradigm for complexity . Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ Chaos theory studies systems that are flowing. DocWeather : The Weather Eye > Weather and Chaos Theory: the Standard Map 19 January 2010 9:53 UTC docweather.com [Source type: FILTERED WITH BAYES]
^ 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. Personal Pages  Chaos Theory Project 19 January 2010 9:53 UTC www.fabalou.com [Source type: FILTERED WITH BAYES]
^{[15]}
.^ 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. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
.^ 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]
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^{[16]} .^ Put another way, chaos is a type of nonlinear behavior emerging along a universal route. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ In full blown chaos, disorder almost displaces order; a system may end up almost any place in an outcome field. CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ 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]
.^ As per the current mathematical theory a chaotic system is defined as showing "sensitivity to initial conditions". Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]
Chaotic dynamics
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]
Here two series of
x and
y values diverge markedly over time from a tiny initial difference.
.^ 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 ndimensional 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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ 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 commonlyused definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:
^{[19]}
 it must be sensitive to initial conditions,
 it must be topologically mixing, and
 its periodic orbits must be dense.
Sensitivity to initial conditions
.^ 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]
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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.
.^ 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]
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ The Butterfly Effect reflects how changes on the small scale, can influence things on the large scale. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ In nonlinear systems small changes or small errors can have big effects. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ The butterfly effect is what this effect came to be known as. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ 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. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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]
.^ 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]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ The conditions of the buckets filling and emptying will no longer be so synchronous as to facilitate just simple rotation; chaos has taken over. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ This approach of aggregating cases across disasters and disciplines has been used to arrive at generalized findings about disaster management. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ That not only ruined any hope of forecasting weather beyond a week or so, but similarly hampered our ability to foresee climate change. Chaos in the Atmosphere 19 January 2010 9:53 UTC www.aip.org [Source type: FILTERED WITH BAYES]
^ 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. Chaos in the Atmosphere 19 January 2010 9:53 UTC www.aip.org [Source type: FILTERED WITH BAYES]
^{[24]}
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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
diverge
where λ is the Lyapunov exponent. The rate of separation can be different for different orientations of the initial separation vector.
.^ This strange phenomena was said to reside in what they called phase space and a whole new element of chaos theory was born. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ But in the long run, some of these trajectories could effectively diverge as if there was onaverage 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 phasespace available to it (Region D). TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
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.
.^ 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
y →
x +
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.
.^ 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 ndimensional 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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ To summarize, the folklore that trajectories issuing forth from neighboring points will diverge onaverage 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 phasespace. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ A concept involved in chaotic systems is fractals. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ It is a mathematically constructed conceptual space where each dimension corresponds to one variable of the system (Kellert, 1993). Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ An example in the world of physics of such a system would be a fluid in turbulent motion (1). Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ 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 onset of the disaster. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ This phenomenon is also known as sensitive dependence on initial conditions and is common to the chaos theory. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ As per the current mathematical theory a chaotic system is defined as showing "sensitivity to initial conditions". Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ If you look at some of the examples giving of dynamical systems, it is clear that some of them are predictable. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [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]
^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ Another system in which sensitive dependence on initial conditions is evident is the flip of a coin. Chaos Theory: A Brief Introduction  IMHO 19 January 2010 9:53 UTC www.imho.com [Source type: FILTERED WITH BAYES]
^ It was a theorem on the sensitive dependence on initial conditions about the frictionless motion of a point on a surface. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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]
However, this example has no topological mixing, and therefore has no chaos.
.^ Repeated application of a simple function causes some of these points to flee toward infinity, while others never wander far from the origin. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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
.^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ (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]
.^ 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]
^ Sensitivity to Initial Conditions Systems functioning in chaotic regimes may show a tendency to be highly sensitive to their initial conditions. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ This established that the system is inherently unpredictable and as a result any slight error in the initial conditions would be amplified rapidly. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]
^{[25]} .^ For example, if one begins with a continuous system, by using a Poincaré surface of section—roughly, a twodimensional 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 onedimensional 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 onetomany or manytomany? 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]}
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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 onaverage 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.
.^ Chaos research maps the geometry of system dynamics in phasespace. CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ It will also depend on the complexity of the dynamic that defines the system. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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 ndimensional 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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
.^ Those points in convergent regions form a Fatou set. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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 selfsimilarity 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]
.^ A Julia set is a set of extraordinary points that separate different basins of attraction. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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 . Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ One plots the boundary between basins of attraction; the other graphs the attractor of the inverse relation. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ A value for c that lies on one of the filaments creates a similarly shaped Julia set, because infinity is the only attractor. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Right now we would like to pause along the route to chaos to offer a mock application of such phase transitions for heuristic purposes. CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ The attractor in figure 5 is derived from a chaotic time series generated by the logistic equation. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ On the other hand, the dissipative chaotic models used to characterize some realworld 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]
.^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ Most have long known that the solar system does not "run with the precision of a Swiss watch." Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ The first such attractor was identified by Edward Lorenz in his meteorology simulations in the early 1960s (Lorenz, 1963). TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ 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). Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ Attractor, Strange: A strange attractor is simply the pattern, in visual form, produced by graphing the behavior of a nonlinear system. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ 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. CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ There are two main types of dynamic systems: discrete and continuous. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ The first such attractor was identified by Edward Lorenz in his meteorology simulations in the early 1960s (Lorenz, 1963). CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ 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]
.^ Systems with dissipative structures are called dissipative systems. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ 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. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ 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
.^ Using The Logistic Map To Track Disaster Organizational Morphogenesis Recall that the organizational states depicted by the logistic diagram take place during a twentyfour hour period. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ Generally, the map of disaster organization morphogenesis defines where an organization or response system is on the logistic diagram. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
Each vertical slice shows the attractor for a specific value of
r. The diagram displays
perioddoubling as
r increases, eventually producing chaos.
.^ For example, if one begins with a continuous system, by using a Poincaré surface of section—roughly, a twodimensional 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 realworld 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]
.^ 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. TITLE 19 January 2010 9:53 UTC critcrim.org [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]
.^ 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 finitetime dynamics for each system because these may not yield any onaverage 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 nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ Which means that if you very slightly change a parameter in an equation or system, very different behaviors can result (29). Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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 ? WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ 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. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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). CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ 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]
.^ Nonlinear systems exhibit three distinct types of behavior over time. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ Chaos is one possible result of the dynamics of nonlinear systems. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ One method for controlling chaos is to alter the parameters of the system. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ Not only is our body a dissipative system, but our ego as well. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
.^ For example, if one begins with a continuous system, by using a Poincaré surface of section—roughly, a twodimensional 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). TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ 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). TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ Dynamical systems are comprised of two elements. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ For example, if one begins with a continuous system, by using a Poincaré surface of section—roughly, a twodimensional 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^{.December 2009" style="whitespace:nowrap;">[citation needed]} Perhaps surprisingly, chaos may occur also in linear systems, provided they are
infinitedimensional.
^ 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 ndimensional 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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ A system exhibiting nonlinear behavior may appear quite random over time, yet studies of chaotic regimes in phasespace reveal underlying patterns. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^{[31]} .^ "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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ 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 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). Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
History
.^ Fractals and modern chaos theory are also linked by the fact that many of the contemporary pacesetting discoveries in their fields were only possible using computers. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ Depicting natural things such as clouds and mountain ranges has been a challenge for computer graphics systems based on everyday geometry. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
) may be recreated through an
Iterated function system (IFS).
The first discoverer of chaos was
Henri Poincaré.
.^ It was from this approach he was able to show that the threebody problem has complicated orbital dynamics, which we now call chaos. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^{[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]} .^ For systems exhibiting SDIC, trajectories starting out in a highly localized region of state space will diverge onaverage 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 onaverage 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 onaverage exponentially diverging trajectories is invalid. Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]
.^ Indeed another great mathematician named Gaston Julia came up with a long memoir developing the theory in a similar way to Fatou. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ Number theory and the study of nonlinear differential equations also give examples of fractal sets. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^{[40]} .^ Except for Smale, who was perhaps the first pure mathematician to study nonlinear dynamics, these studies were all directly inspired by physics: the threebody 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. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
^ Except for Smale, these studies were all directly inspired by physics: the threebody 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. What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ 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]} .^ 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. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
 What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ Their findings supported the theory that chaotic attractors cause fluid turbulence. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Chaos theory is a way to analyze such 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]
^ 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 ndimensional 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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
.^ 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. CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Chaos Theory: A science which deals with the complex harmonies and disharmonies exhibited by natural and social systems. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ The arrival of digital computers has accelerated the pace of development in the field of chaos. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Fractals and modern chaos theory are also linked by the fact that many of the contemporary pacesetting discoveries in their fields were only possible using computers. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ "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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
.^ 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. CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ Electronic computers made these repeated calculations practical. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
 What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ 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. Chaos Theory and Market Reality, Part One 19 January 2010 9:53 UTC www.rbtrading.com [Source type: FILTERED WITH BAYES]
Turbulence in the
tip vortex from an
airplane wing.
.^ Chaos theory is a way to analyze such 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]
^ For this reason, we can say that the transition from laminar flow to turbulence is an initiation of selforganization it is the creation of order from chaos. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ 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. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
David Ruelle and
Floris Takens later predicted, against Landau, that
fluid turbulence could develop through a
strange attractor, a main concept of chaos theory.
.^ In 1960 he was working on the dilemma of weather prediction. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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]
^ The first such attractor was identified by Edward Lorenz in his meteorology simulations in the early 1960s (Lorenz, 1963). CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
 What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ To save time, he started in the middle of the sequence, instead of the beginning. Chaos Theory: A Brief Introduction  IMHO 19 January 2010 9:53 UTC www.imho.com [Source type: FILTERED WITH BAYES]
^ He started in the middle of the first number sequence. Math G Projects, Mission College, Santa Clara, Chaos 19 January 2010 9:53 UTC www.missioncollege.org [Source type: FILTERED WITH BAYES]
.^ 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. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
 What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
 What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ Using real time disaster response data or a simulated exercise, test Koehler's and Comfort's hypothesis about how disaster wide networks emerge . WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ 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. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
^ To his surprise the weather that the machine began to predict was completely different from the weather calculated before. What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ Lorenz tracked this down to the computer printout. What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
The computer worked with 6digit precision, but the printout rounded variables off to a 3digit 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. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
 What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ 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. GameSpy: Splinter Cell Chaos Theory  Page 1 19 January 2010 9:53 UTC pc.gamespy.com [Source type: General]
^ The butterfly effect can be better understood by examining the two different time series in figure 4. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ However Lorenz had discovered that small changes in initial conditions produced large changes in the longterm outcome.The term chaos as used in mathematics was coined by the applied mathematician James A. Yorke. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
^ 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. Chaos Theory  From the New Universe 19 January 2010 9:53 UTC www.newuniverse.co.uk [Source type: FILTERED WITH BAYES]
^ However Lorenz had discovered that small changes in initial conditions produced large changes in the longterm outcome. What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^{[42]} .^ Yet there was little reason to doubt that precise predictability could, in principle, be achieved. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ From that idea Lorenz stated that it is impossible to predict the weather accurately. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ Mandelbrot has found order in places where others before him saw only chaos. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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 ? WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^{[43]} .^ Number theory and the study of nonlinear differential equations also give examples of fractal sets. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^{[44]} .^ 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. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ Chaos theory thus adds understanding and support to the logic of many new management ideas. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ Both higher prices and higher taxes are required thus reducing competitive advantages and neglecting essential common social needs. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^{[45]}^{[46]} In 1967, he published "
How long is the coast of Britain? .^ The selfsimilar 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 computergenerated fractal. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ An important geometric implication of selfsimilarity 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]} .^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ Thus the Koch snowflake is an infinitely long curve that encloses a finite area. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ As can be observed, the fractal curve is selfsimilar. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ The Koch snowflake clearly illustrates the concept of fractal dimension. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ Chaos theory has also given us fractal geometry, the visual representation of chaos theory. EXPLORIT Science Center  "Chaos Theory" 19 January 2010 9:53 UTC www.explorit.org [Source type: FILTERED WITH BAYES]
^ Classical physics, even including the advanced ramifications of quantum theory, deals primarily with ordered systems, but the natural world exhibits a tendency toward disorder. CHAOS THEORY, 19 January 2010 9:53 UTC www.history.com [Source type: General]
^ Still since it is describable by deterministic equations, chaos theory supports a strictly deterministic philosophy of nature, although within subtle epistemic limits.
.^ 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]}
.^ 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). Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
^ I will return to the question of using mathematical models to represent realworld systems throughout this article. Chaos (Stanford Encyclopedia of Philosophy) 19 January 2010 9:53 UTC plato.stanford.edu [Source type: Academic]
.^ Several years later May found an instance of chaos in the iterative map (logistic map) that was used to study population dynamics in biology. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Immediately following a disaster organizations occupy different places on the logistic map. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]} .^ Most forms that occur in nature are fractals. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Fractal geometry has even been applied to computeraided music composition. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Corollary to such work is the need to cultivate a mathematics and a geometry with which to study such dynamics. TITLE 19 January 2010 9:53 UTC critcrim.org [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]
^ Application of chaos theory to family interaction . Readings: KoopmansFamily and Chaos Theory 19 January 2010 9:53 UTC chiron.valdosta.edu [Source type: Academic]
^ Chaos Theory is no different. Tom Clancy�s Splinter Cell: Chaos Theory On GameVortex.com 19 January 2010 9:53 UTC www.psillustrated.com [Source type: General]
.^ 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. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ For this reason, we can say that the transition from laminar flow to turbulence is an initiation of selforganization it is the creation of order from chaos. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ 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]
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]}
.^ 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]
^ 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. 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 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. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ 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 onetoone 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]} .^ 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 nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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]
.^ The capacity to develop a selforganizing response involves the whole system and varying social time from top to bottom. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ How various agencies link together, their social times, their ability to selforganize and mobilize resources are all directly related to how quickly the response emerges. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ This suggests, that attention to social time entrainment and self organizing processeswhat is necessary within a particular context to bring about a rapidly developing overall responseis critical, rather than attention to quantitative objectives alone. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ This property is known as scale invariance . Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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. 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: A science which deals with the complex harmonies and disharmonies exhibited by natural and social systems. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ This property is known as scale invariance . Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ The surfaces of certain cheeses and the random distribution of the stars in the sky display the property known as statistical selfsimilarity. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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]
.^ 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 "lowrisk". CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]
.^ The model developed for this article is one of a twoadversary 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. CHAOS THEORY, 19 January 2010 9:53 UTC www.history.com [Source type: General]
^ In physics, though, chaotic systems are classical in scale and thus subsumable in principle under classical mechanics with its deterministic laws of motion.
.^ Yet, what is it that makes chaos theory so fascinating? Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Chaos theory isn't new to astronomers. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ Computergenerated images applying Chaos Theory principles. Chaos Theory (Lorenz Poincar�) 19 January 2010 9:53 UTC www.12manage.com [Source type: FILTERED WITH BAYES]
.^ A nonlinear system is a system that is not in the first degree. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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 nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
.^ Chaos theory has many new concepts with which to grasp its ideas. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ 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. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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. What is Chaos Theory?  Yahoo! Answers 19 January 2010 9:53 UTC answers.yahoo.com [Source type: FILTERED WITH BAYES]
^ 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. Chaos Homepage 19 January 2010 9:53 UTC www.zeuscat.com [Source type: FILTERED WITH BAYES]
.^ 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) evenlyspread
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
.^ Time series data may be nonlinear but not chaotic. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ [Review of the article Can the analytic techniques of nonlinear dynamics distinguish periodic, random and chaotic signals? Bibliography of Chaos Theory 19 January 2010 9:53 UTC www.southernct.edu [Source type: Academic]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
.^ With this allpurpose 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. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ 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]
.^ Chaos thus looks like random behavior but is really unstable behavior over time that stays within clear boundaries. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ But even if some of the changes are marginal, this is still the most entertaining, most wellrounded 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]}
.^ 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. Chaos Theory: A Brief Introduction  IMHO 19 January 2010 9:53 UTC www.imho.com [Source type: FILTERED WITH BAYES]
^ 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. CHAOS THEORY 19 January 2010 9:53 UTC uwacadweb.uwyo.edu [Source type: FILTERED WITH BAYES]
 TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^{[59]}^{[60]} Thus, given a time series to test for determinism, one can:
 pick a test state;
 search the time series for a similar or 'nearby' state; and
 compare their respective time evolutions.
.^ We see that a small difference, a small error, or a small change can have very novel, unexpected and even explosive effects over time. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]
.^ 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. Order/Chaos Relationship 19 January 2010 9:53 UTC www.schuelers.com [Source type: FILTERED WITH BAYES]
^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
A stochastic system will have a randomly distributed error.
^{[61]}
.^ So analysts need to be careful not to call all "messy" looking time series data as chaotic. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ You can find almost all the nike series there, in huge collection and varies colorways. The American Spectator : Chaos Theory 19 January 2010 9:53 UTC spectator.org [Source type: FILTERED WITH BAYES]
^ 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]
).
.^ Any state of the system at a moment in time is represented as a point in phase space. Personal Pages  Chaos Theory Project 19 January 2010 9:53 UTC www.fabalou.com [Source type: FILTERED WITH BAYES]
^ A phase space diagram is a history of the changing variables of the system. Personal Pages  Chaos Theory Project 19 January 2010 9:53 UTC www.fabalou.com [Source type: FILTERED WITH BAYES]
^ 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]} .^ Like the flow in the rotating dishpan, the system had at least two states that it could flip between for no obvious reason. Chaos in the Atmosphere 19 January 2010 9:53 UTC www.aip.org [Source type: FILTERED WITH BAYES]
^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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.
.^ As such, this defining characteristic could be applied to both mathematical models and realworld 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]
.^ 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. WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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 ? WHAT DISASTER RESPONSE MANAGEMENT CAN LEARN FROM CHAOS THEORY 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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]
.^ 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]
.^ For many systems, the addition of the nth doubling will cause it to transform into nonlinearity. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ Chaos theory is the mathematics of studying such nonlinear, 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 nonlinear states of such dynamical systems. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
.^ Chaos theory is the mathematics of studying such nonlinear, 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 nonlinear states of such dynamical systems. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ According to respected authorities, stock markets are nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate the deterministic part risk failure.
.^ For many systems, the addition of the nth doubling will cause it to transform into nonlinearity. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ Chaos theory is the mathematics of studying such nonlinear, 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 nonlinear states of such dynamical systems. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^{[63]} .^ 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. CHAOS THEORY AND DISASTER RESPONSE MANAGEMENT 19 January 2010 9:53 UTC www.library.ca.gov [Source type: FILTERED WITH BAYES]
^ 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 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. TITLE 19 January 2010 9:53 UTC critcrim.org [Source type: FILTERED WITH BAYES]
^ Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
^ 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
 Examples of chaotic systems




References
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 ^ Kellert, p. 56.
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 ^ 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.
 ^ Applying Chaos Theory to Embedded Applications
 ^ HristuVarsakelis, 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.
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 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]
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 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 nonlinear, dynamic systems. Chaos Theory 19 January 2010 9:53 UTC www.emayzine.com [Source type: FILTERED WITH BAYES]
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 ^ 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.
 ^ Benoît Mandelbrot (5 May 1967) "How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension," Science, Vol. 156, No. 3775, pages 636–638.
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 ^ R.L. Ives (10 October 1958) "Neon oscillator rings," Electronics, vol. 31, pages 108–115.
 ^ 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).
 ^ Ralph H. Abraham and Yoshisuke Ueda, eds., The Chaos AvantGarde: Memoirs of the Early Days of Chaos Theory (Singapore: World Scientific Publishing Co., 2001). See Chapters 3 and 4.
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 ^ Per Bak, Chao Tang, and Kurt Wiesenfeld, "Selforganized 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.nslijgenetics.org/wli/1fnoise/1fnoise_square.html . See especially: Lasse Laurson, Mikko J. Alava, and Stefano Zapperi, "Letter: Power spectra of selforganized critical sand piles," Journal of Statistical Mechanics: Theory and Experiment, 0511, L001 (15 September 2005).
 ^ F. Omori (1894) "On the aftershocks of earthquakes," Journal of the College of Science, Imperial University of Tokyo, vol. 7, pages 111–200.
 ^ [3]The Independent, Fri 29 August 2003
 ^ ^{a} ^{b} Provenzale A. et al.: "Distinguishing between lowdimensional dynamics and randomness in measured timeseries", in: Physica D, 58:31–49, 1992
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 ^ Kyrtsou, C., (2008). Reexamining the sources of heteroskedasticity: the paradigm of noisy chaotic models, Physica A, 387, pp. 6785–6789.
 ^ 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
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