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# Topic: Dynamical systems and chaos theory

###### In the News (Sun 21 Jul 19)

 Dynamical systems and chaos theory: Definition and Links by Encyclopedian.com Dynamical systems and chaos theory is the branch of mathematics that deals with the long-term qualitative behavior of dynamical systems. An important goal is to describe the fixed points, or steady states of a given dynamical systems; these are values of the variable which won't change over time. The branch of dynamical systems which deals with the clean definition and investigation of chaos is called chaos theory. www.encyclopedian.com /dy/Dynamical-systems.html   (236 words)

 Dynamical system Summary Newton's theories of dynamics (laws of motion) had a profound effect on the world of science, particularly with the realization that the physical laws of the Earth were the same as the laws of the planets. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood. The PoincarÃ© recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms. www.bookrags.com /Dynamical_system   (4365 words)

 Chaos FAQ   (Site not responding. Last check: 2007-10-03) Dynamical systems theory is the branch of mathematics devoted to the motions of systems which evolve according to simple rules. Chaos theory is a further development of dynamical systems theory which focusses on highly complex motions called chaotic motions. As chaos theory is a new branch of math, it is relatively independent of the main topics of the trditional progrtam. www.vismath.org /faq/chaosfaq.html   (619 words)

 Theoretical Reseach - UMD Chaos Group Unlike nonchaotic cases, where the system settles into an equilibrium or regular oscillatory mode, a system whose evolution is described by a chaotic attractor exhibits many of the properties of a random process. Systems that entered a regime in which they oscillated irregularly were equated with random processes, and therefore considered unpredictable except in a statistical way. Among recent results in quantum chaos is a prediction relating the chaos in the classical problem to the statistics of energy-level spacings in the semiclassical quantum regime. www-chaos.umd.edu /research.html   (1830 words)

 SystemsTheoryandChaos The systems regarded as typical of the nature of the universe were those where explicit mathematical prediction was possible, such as the pendulum and the system consisting of a single planet in orbit round a star. In mathematical terms, the emergence of the new structure of convection is loosely analogous to the onset of dynamic instability. An inspiring tour through the social implications of thinking in a dynamical systems manner about the society we live in, which she believes is leading to a major shift in human culture. www.greenspirit.org.uk /resources/SystemsTheoryandChaos.htm   (1916 words)

 [No title] Chaos theory is one of a set of approaches to study nonlinear phenomena. Agent-based models are closely related to several other computational systems that illustrate self-organization dynamics such cellular automata, genetic algorithms, and the spin-glass models that formed the basis of NK or Rugged Landscape models of self-organizing behavior. It would be correct to call chaos theory in psychology a new paradigm in psychological thought: Nonlinear theory introduces new concepts to psychology for understanding change, new questions that can be asked, and offers new explanations for phenomena. www.societyforchaostheory.org /tutorials   (2270 words)

 Chaos   (Site not responding. Last check: 2007-10-03) Chaos is qualitative in that it seeks to know the general character of a system's long-term behavior, rather than seeking numerical predictions about a future state. Finally, a dynamic system is a simplified model for the time-varying behavior of an actual system. Edward Lorenz would stretch the definition of chaos to include phenomena that are slightly random, provided that their much greater apparent randomness is not a by-product of their slight true randomness. www.exploratorium.edu /complexity/lexicon/chaos.html   (475 words)

 Open Questions: Chaos Theory and Dynamical Systems Mainly consists of Chaos Without the Math -- a collection of essays on the history of chaos theory, instability, strange attractors, phase transitions, deep chaos, and self-organization. Dynamical chaos -- the appearance of seemingly random motion in a deterministic dynamical system -- can occur even in systems with few degrees of freedom. The book's eleven chapters deal with topics such as chaos, fractals, dynamical systems, and applications to meteorology, biology, and fluid flow by focusing on the individuals and groups that have made the main contributions. www.openquestions.com /oq-ma002.htm   (2261 words)

 Dynamical system - Wikipedia, the free encyclopedia The Lorenz attractor is an example of a non-linear dynamical system. The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. en.wikipedia.org /wiki/Dynamical_systems_and_chaos_theory   (3527 words)

 The Math Forum - Math Library - Dynamical Systems Chaos, Dynamic Systems, and Fractal Geometry - Tyeler Quentmeyer, ThinkQuest 1996 Chaos Theory and Fractal Geometry - Kelleen Farrell A database of around 50,000 planar dynamical systems of the type described by the formula: x = f(x,y), y = g(x,y), where f and g are algebraic functions containing constants and the four operations. mathforum.org /library/topics/dynamical_systems   (2043 words)

 Chaos - Interactive Mathematics Online   (Site not responding. Last check: 2007-10-03) Chaos is an advanced field of mathematics that involves the study of dynamical systems, that is, systems in motion. One example of a chaotic dynamical system is the motion of the stars, planets, and galaxies, which mathematicians and scientists have been trying to understand for centuries. The study of chaotic dynamical systems has many applications for the 'real world.' Think of any mathematical system that changes over time, such as the weather, the stock market, or the genetic distribution of a population. library.thinkquest.org /2647/chaos/chaos.htm   (312 words)

 Studio5670Wiki - Yan Zheng Dynamical Systems and Chaos Theory: Simple dynamical system can exhibit a completely unpredictable behavior, which might seem to be random. It should be a space together with a transformation of space, such as solar system transforming over the time according to the equations of the celestial mechanics. The Dynamical System should be a process in which a function value changes over time according to a rule that is defined in terms of the function’s current value which called Continuous Dynamical System. www.uta.edu /architecture/wiki/studio5670/index.php/YanZheng?version=1   (111 words)

 Chaos Theory in Mathematical Modeling II Chaos theory is very much a 20th century development, but the man who probably best deserves the title "Father of Chaos Theory" was a great French mathematician of the 19th century named Henri Poincaré. As discussed on the Dynamical Systems page, Isaac Newton had given the world what seemed to be the final word on how the solar system worked. Prediction becomes impossible." This is the statement which gives Poincare the claim to the title "Father of Chaos Theory." This is the first known published statement of the property now known as "sensitivity to initial conditions", which is one of the defining properties of a chaotic dynamical system. home.earthlink.net /~srrobin/chaosb.html   (937 words)

 Introduction to the Modern Theory of Dynamical Systems Meines Erachtens stellt Katok und Hasselblatts "Introduction to the modern theory of dynamical systems" eine äußerst wertvolle Bereicherung der Literatur über die Theorie dynamischer Systeme dar, und ich kann das Buch jedem uneingeschränkt empfehlen, der diese Theorie in Lehre oder Forschung behandelt oder anwendet. The theory of dynamical systems is a major mathematical discipline closely intertwined with all main areas of mathematics. For hyperbolic systems there are structural stability, theory of equilibrium (Gibbs) measures, and asymptotic distribution of periodic orbits, in low-dimensional dynamical systems classical Poincaré-Denjoy theory, and Poincaré-Bendixson theories are presented as well as more recent developments, including the theory of twist maps, interval exchange transformations and noninvertible interval maps. www.tufts.edu /~bhasselb/thebook.html   (1062 words)

 For Students   (Site not responding. Last check: 2007-10-03) Dynamical systems are mathematical models describing the evolution of systems in terms of equation of motion and initial values. Dynamical systems theory is also finding an increasing number of applications in social sciences as mathematical economy and finance. Here is a mathematica notebook analysing the dynamics of a two dimensional system modelling two species rabbits and sheeps competing for the same food resources. www.math.helsinki.fi /mathphys/dscourse-06.html   (856 words)

 37: Dynamical systems and ergodic theory Dynamical systems is the study of iteration of functions from a space to itself -- in discrete repetitions or in a continuous flow of time. Rather we have a loose collection of tools and techniques, many of them from the classical theory of differential equations, and a guiding global geometrical viewpoint that originated with Poincaré over a hundred years ago and that was further developed by G D Birkhoff, D V Anosov and S Smale and other mathematicians. In particular, the former subfield 58F (Dynamical systems) is one of the largest 3-digit subfields in the MR database (and contains two(!) of the largest 5-digit areas -- 58F07, Completely integrable systems, and 58F13 Strange attractors; chaos). www.math.niu.edu /~rusin/known-math/index/37-XX.html   (714 words)

 Dynamical Systems (Including Chaos) Chaotic systems have many fascinating properties, and there is a good deal of evidence that much of nature is chaotic; the solar system, for instance. Jacky Cresson and Sébastien Darses, "Stochastic embedding of dynamical systems", math.PR/0509713 [112pp. In some cases, for example when studying the long term behaviour of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modelled or even known. cscs.umich.edu /~crshalizi/notebooks/chaos.html   (3360 words)

 Preface Studies of nonlinear systems are truly interdisciplinary, ranging from experimental analyses of the rhythms of the human heart and brain to attempts at weather prediction. Some of these tools are mathematical, such as the application of symbolic dynamics to nonlinear equations; some are experimental, such as the necessary circuit elements required to construct an experimental surface of section; and some are computational, such as the algorithms needed for calculating fractal dimensions from an experimental time series. Many experiments in nonlinear dynamics are individual or small group projects in which it is common for a student to follow an experiment from conception to completion in an academic year. www.drchaos.net /drchaos/Book/node2.html   (1327 words)

 Devaney Books Half the book is devoted to one-dimensional dynamics, the remainder equally split between higher dimensional dynamics and complex dynamics. This is a series of four paperback books on dynamical systems for high school students and their teachers. This is the Proceedings of a conference on Complex Dynamics held at Snowbird, Utah, June 13-17, 2004. math.bu.edu /people/bob/books.html   (577 words)

 Dynamical Systems in Mathematical Modeling II   (Site not responding. Last check: 2007-10-03) Thus, the solar system is a dynamical system; the United States economy is a dynamical system; the weather is a dynamical system; the human heart is a dynamical system. Mathematically speaking, a dynamical system is a system whose behavior at a given time depends, in some sense, on its behavior at one or more previous times. Furthermore, it is the task of the mathematical modeler to come up with a mathematical construct, a model, that will describe this relationship between current and past states of the system so that predictions about the future course of events for the system may be made with some degree of accuracy. home.earthlink.net /~srrobin/dynsysb.html   (770 words)

 American Civilization as a Dynamical System It is these systems, the behavior of which we wish to understand, that are not 'explainable' by the simple notions of a direct 'cause' and 'effect' and the application of Occam's Razor. If the behavior of such a social system produces a periodic stable equilibrium, there is 'predictive' power of a general nature for future 'states' of the system. That structure is a miniature microcosm of the overall behavior of the system in the periodic region. www.newtotalitarians.com /AmericanCivDynamicalSystem.html   (6481 words)

 Amazon.com: Chaos in Dynamical Systems: Books: Edward Ott   (Site not responding. Last check: 2007-10-03) It is intended to serve both as a graduate course text for science and engineering students, and as a reference and introduction to the subject for researchers. This complicated behaviour, called chaos, occurs so frequently that it has become important for workers in many disciplines to have a good grasp of the fundamentals and basic tools of the emerging science of chaotic dynamics. The emphasis is not on time-series analysis or nonlinear systems, but chaos in "physical" systems (in the sense of applications in physics). www.amazon.com /Chaos-Dynamical-Systems-Edward-Ott/dp/0521437997   (1245 words)

 Theory Group, Other Activities The purpose of the theoretical calculations is to gain better understanding of some of the quantum mechanical phenomena that are typical to systems of many particles. Jan FrÃ¸yland is working on the theory of dynamical systems, neural networks, time series analysis and self organized processes. In dynamical systems theory mainly coupled, non-linear oscillators have been studied. www.fys.uio.no /teori/forskning/other.html   (472 words)

 number theory and dynamical systems An indirect link between dynamical systems and number theory is provided by the theory of dynamical zeta functions. Verjovsky, "Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold", Dyamical Systems. However, in the dynamical systems theory to chaos the starting point is the enumeration of asymptotic motions of a dynamical system, and through this enumeration number theory enters and comes to play a central role. secamlocal.ex.ac.uk /~mwatkins/zeta/dynamicalNT.htm   (6443 words)

 620-341 Dynamical Systems & Chaos This subject introduces the basic concepts and recent developments in the fields of dynamical systems and chaos, including stability of equilibria and renormalisation theory of transitions to chaos. Students should develop the ability to analyse simple nonlinear discrete and continuous dynamical systems, and to chart parameter regions of stability, periodicity and chaos. This subject demonstrates the power as well as the limitations of dynamical systems theory and chaos applied to realistic complex systems. www.unimelb.edu.au /HB/2002/subjects/620-341.html   (204 words)

 Amazon.com Message   (Site not responding. Last check: 2007-10-03) A coherent framework is presented for understanding the dynamics of piecewise-smooth and hybrid systems. SDSs are a class of discrete dynamical systems which are a significant generalization of cellular automata and provide a new general theory of discrete computer simulations. Fascinating and authoritative, Chaos and Fractals: New Frontiers of Science is a truly remarkable book that documents recent discoveries in chaos theory with plenty of mathematical detail, but without alienating the general reader. www.amazon.com /exec/obidos/tg/feature/-/58482   (394 words)

 Dynamical Systems and Chaos We will concentrate on the simplest dynamical systems which can exhibit so-called chaotic behaviour -- the discrete dynamical systems which arise from iterations of real or complex valued functions. The perspective taken in dynamical systems is to attempt to understand the qualitative behaviour of a whole system or classes of systems rather than writing down particular explicit solutions. Computers can be an effective tool for "experimentally" discovering properties of dynamical systems, especially discrete ones, and can lead to theoretical discoveries too. www.math.jhu.edu /~mhaskin/teaching/dynamics2002/chaos.html   (808 words)

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