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	Chaos theory - Wikipedia, the free encyclopedia
		Sensitivity to initial conditions means that two points in such a system may move in vastly different trajectories in their phase space even if the difference in their initial configurations is very small.
		The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly.
		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.
	en.wikipedia.org /wiki/Chaos_theory (2011 words)

	Chaos theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		Systems that exhibit mathematical chaos are (additional info and facts about deterministic) deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder.
		A non-chaotic system is generally better understood than a chaotic system and therefore perhaps less interesting as a plot device in science fiction.
		Many simple systems can also produce chaos without relying on (An equation containing differentials of a function) differential equations, such as the (additional info and facts about logistic map) logistic map, which is a difference equation ((additional info and facts about recurrence relation) recurrence relation) that describes population growth over time.
	www.absoluteastronomy.com /encyclopedia/c/ch/chaos_theory.htm (2087 words)

	Chaos Theory and Fractals (Site not responding. Last check: 2007-10-18)
		Behavior in chaotic systems is aperiodic, meaning that no variable describing the state of the system undergoes a regular repetition of values.
		One of the most interesting issues in the study of chaotic systems is whether or not the presence of chaos may actually produce ordered structures and patterns on a larger scale.
		Chaos is also found in systems as complex as electric circuits, measles outbreaks, lasers, clashing gears, heart rhythms, electrical brain activity, circadian rhythms, fluids, animal populations, and chemical reactions, and in systems as simple as the pendulum.
	www.mathjmendl.org /chaos (1990 words)

	Mastering Chaos
		The chaotic attractor is the manifestation of the fixed parameters and equations that determine the values of the dynamic variables.
		Chaotic motions, however, can often be more conveniently visualized in "state space," a plot of the history of the changing variables, which are typically the position and velocity of the object.
		In a rigorous sense, the chaotic attractor is an ensemble of unstable periodic orbits.
	www.fortunecity.com /emachines/e11/86/mastring.html (5534 words)

	[No title]
		These circuit based systems have not yet been demonstrated to have the security of systems such as DES or RSA, but they are inexpensive to produce, and may be useful in analog applications such as portable phones which currently suffer greatly from theft of services (due to their weak or nonexistent encryption algorithms).
		Intermittency Chaotic systems can exhibit a phenomenon called intermittency where the observed behavior is apparently periodic for a long time (compared to the typical time period of the system) and abruptly switches to chaotic behavior, or vice versa (Hilborn, 1994).
		A straightforward variant of this would be to have a system of N independent logistic maps (the exact value of N depending on the length of the key), using N-1 of them to generate pseudorandom sequences, and one to decide which one to pick the output from at each timestep.
	www.cs.wisc.edu /~yetkin/code/crypto/crypto_paper.doc (1750 words)

	M.Schwartz: Chaotic Systems and Blake's Mythology
		Chaotic systems are therefore fundamentally unpredictable yet ordered; this is the definition of "chaotic" order, chaos theory thus demonstrating that there is a "deeper" order within the apparent randomness of a chaotic system.
		The "chaotic" process of this system is dependent on the its interaction with its surroundings; this influence on the ensuing chaos of a system is precisely what nonlinear parameters try to describe.
		As with other chaotic systems, dissipative structures derive an internal order from their instability, and it is by means of their constant lapsing into disorder and rebounding into organization that they "evolve," proceeding toward some unimaginably complicated yet structured end, despite the efforts of the supposedly-dominant force of entropy.
	prometheus.cc.emory.edu /panels/3A/M.Schwartz.html (3069 words)

	Theoretical Reseach - UMD Chaos Group
		In chaotic transients one observes that typical initial conditions initially behave in an apparently chaotic manner for a possibly long time, but, after a while, then rapidly move off to some other region of phase space, perhaps asymptotically approaching a nonchaotic attractor.
		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.
	www-chaos.umd.edu /research.html (1830 words)

	Ordinary Differential Equations: Chaotic Systems
		As the name indicates they are systems that are modelled using discrete time scheme where, in general, the current state of the system depends upon state of the variables in the previous state.
		An example of a dynamical system in which time is a continuous variable is a system of N first-order,ordinary differential equations, expressed as,
		To reveal the connection between the chaos generating mechanism for the case of Logistic map and that of the continuous time Rossler system, we use the technique of Poincaré surface of section (taken when the phase variable x goes through a maxima) and plot the current maxima against the previous maxima.
	chaos.phy.ohiou.edu /~thomas/chaos/ode.html (1287 words)

	Abstracts
		Furthermore, no chaotic orbits lying on a torus were observed, suggesting that, in most cases, at least in the case of this system, orbits do not become chaotic before their tori are destroyed.
		The phenomenology of this bifurcation is that the scattering is chaotic on both sides of the bifurcation, but, as the system parameter pa sses through the critical value, an infinite number of periodic orbits are destroyed and replaced by a new infinite class of periodic orbits.
		The sensitivity of chaotic systems to small perturbations is used to direct trajectories to a small neighborhood of stationary states of three-dimensional chaotic flows.
	www-chaos.umd.edu /publications/abstracts.html (4733 words)

	Fractal and Chaotic Dynamics in the Brain/1
		In a time-varying system, chaos may become established by three principal routes involving a (possibly infinite) sequence of bifurcations of the attractor, intermittent disruption of a periodicity, or the topological breakup of a surface, such as a torus, representing several linked oscillations.
		In a chaotic system in one or more variables, sensitive dependence requires at least one of the Liapunov exponents to be greater than 1, thus resulting in exponential separation of trajectories.
		It is important to be able to distinguish chaotic systems from systems which may have both multiple periodicity and a degree of external noise or stochastic behavior.
	www.dhushara.com /book/paps/chaos/bchaos1.htm (7208 words)

	Chaos
		Thus chaos theory is, very generally, the study of forever changing complex systems based on mathematical concepts of recursion, whether in the form of a recursive process or a set of differential equations modeling a physical system.
		By chaotic, we mean that the particle's location, while definitely in the attractor, might as well be randomly placed there.
		After a while, though, he found that while the momentary behavior of the particle was chaotic, the general pattern of an attractor appeared.
	www.crystalinks.com /chaos.html (768 words)

	Chaotic Systems Team
		Due to the overlap of spectra of noise and chaotic signal, only a small part of the additive noise can be removed by conventional linear filtering in chaotic communications sytems.
		However, the knowledge of dynamics of chaotic system used to generate the chaotic carrier makes possible the application of standard optimization techniques, well-known from the adaptive signal processing, to perform the noise reduction.
		To illustrate the operation of the proposed system, the most important signals and the operation of noise cleaning block can be observed by pushing the red buttons in the figure.
	www.mit.bme.hu /research/chaos/cleaning (897 words)

	Chaotic Dynamical Systems
		In Dynamical Systems, an object moves according to a rule.
		Depending on the rule motion, the object may move in a regular fashion or in a chaotic fashion.
		The user can vary the rules of motion to produce either a regular pattern, a chaotic pattern or a pattern that has a mixture of regular and chaotic behaviour.
	serendip.brynmawr.edu /chaos (253 words)

	Speedy Chaos Control
		For a chaotic circuit, the solution is to apply a voltage pulse during each cycle, with an amplitude that depends on the details of the chaotic behavior.
		The team's algorithm is to apply a constant voltage whenever the state of the chaotic circuit is inside of a small, predetermined rectangular "window" in phase space.
		The window and voltage value are carefully chosen so that on each cycle the pulse pushes the system's state a bit closer to the corner of the window which is closest to the center of the orbits.
	focus.aps.org /story/v4/st14 (635 words)

	Proposal for College of Arts and Sciences (Site not responding. Last check: 2007-10-18)
		This framework has demonstrated that even systems with a few degrees of freedom can produce extremely complex behaviors, which are unpredictable in the long time and are exponentially sensitive to external perturbations.
		This is the appeal of chaotic systems, which on one hand resemble noisy systems with board frequency spectrum, and on the other hand affords deterministic analysis and control.
		In particular, this theoretical grant aims to examine the metamorphosis of the collection of the unstable periodic states as the coupling between two chaotic systems is varied.
	complex.gmu.edu /people/paso/so_gra.html (857 words)

	ipedia.com: Chaos theory Article (Site not responding. Last check: 2007-10-18)
		In mathematics and physics, chaos theory deals with the behaviour of certain nonlinear dynamical systems that exhibit the phenomenon known as chaos, most famously characterised by sensitivity to initi...
		An example of such sensitivity is the well-known butterfly effect, whereby the flapping of a butterfly's wings produces tiny changes in the atmosphere which over the course of time cause it to diverge from what it would have been and potentially cause something as dramatic as a tornado to occur.
		Many simple systems can also produce chaos without relying on partial differential equations, such as the logistic equation, which describes population growth over time.
	www.ipedia.com /chaos_theory.html (1359 words)

	MAPS: [Fwd: Chaotic systems] (Site not responding. Last check: 2007-10-18)
		With a chaotic > system, though, a slight change in initial condition makes > a large change the system behavior, much like what would > happen if you balanced the ball on a knife edge.
		For chaotic systems, > it's not only the initial condition change that produces a large > behavior change, but the condition at each successive time t.
		With a chaotic > system, you'll never be able to measure sensitively enough, > and thus the "in principle unpredictabilty".
	www.maps.org /pipermail/maps_forum/1998-June/000918.html (555 words)

	Advancements Leading to the Discovery of Predictive Deduction
		It was originally intended to model systems of fluid flow, such as curling smoke, but scientists have since discovered that it's also of primary importance in modeling the weather, the stock market, biological systems, and evolution.
		For the same reason that experimentation would be ineffective on random systems (to the extent that the system was random), it's ineffective on chaotic ones.
		A dripping faucet, for example, is a chaotic system composed of parts (pressure, viscosity, etc.) that are individually non-chaotic.
	philosophy.wisc.edu /lang/pd/pd21.htm (1287 words)

	Emergence of Chaotic Systems
		Research at Los Alamos National Laboratory and a number of universities led to an understanding of the emergence of chaotic behavior in systems thought to be deterministic, a concept that has gained widespread acceptance in almost every discipline of science and mathematics.
		Chaotic systems appear in many natural or engineered contexts.
		In the early 1980s, few scientists believed in deterministic chaos; today, virtually all branches of science and engineering interpret nonrandom dynamics in the language of chaos, and all accept that chaos is an important advance in understanding of such systems.
	www.er.doe.gov /Sub/Accomplishments/Decades_Discovery/65.html (391 words)

	USS Clueless - Economic chaos
		Chaos theory is fascinating to me. As a systems engineer, I'm well aware of how complex they can be, and how difficult it can be to understand how a complex system will respond to a particular change in the operating environment.
		That particular system doesn't respond quite like we initially think it would, but at least the feedback mechanism itself is pretty straightforward (unless essential components of it are damaged or missing).
		Some systems naturally oscillate, and some systems are subject to long term alterations in behavior which seem to be permanent and irreversible which are not induced directly by external changes.
	denbeste.nu /cd_log_entries/2004/06/Economicchaos.shtml (4488 words)

	Chaos Homepage
		Chaos theory predicts that complex nonlinear systems are inherently unpredictable--but, at the same time, chaos theory also insures that often, the way to express such an unpredictable system lies not in exact equations, but in representations of the behavior of a system--in plots of strange attractors or in fractals.
		Systems of dynamic equations have been used to model everything from population growth to epidemics to arrhythmic heart palpitations.
		In fact, almost any chaotic system can be readily modeled--the stock market provides trends which can be analyzed with strange attractors more readily than with conventional explicit equations; a dripping faucet seems random to the untrained ear, but when plotted as a strange attractor, reveals an eerie order unexpected by conventional means.
	www.zeuscat.com /andrew/chaos/chaos.html (1678 words)

	RedOrbit - Science - Noise Brings Order to Chaotic Systems (Site not responding. Last check: 2007-10-18)
		Changsong Zhou and a group of physicists at the University of Potsdam, Germany, are studying chaotic systems, known as excitable media.
		The firing of neurons in the brain is an example of such a system, as is the growth and receding of blooms of plankton in the sea.
		Such systems do not become excited by small signals but if they are stimulated above a threshold amount, then they give it their all: neurons fire and plankton blooms.
	www.redorbit.com /news/display?id=123795 (518 words)

	Chaotic Systems
		The trick, in practice, of predicting the path of something or the future course of an evolving system described by a complex algorithm comes in the sensitivity of the algorithm to small changes in the initial point.
		The importance of studying chaotic behavior lies in the fact that most systems encountered in the real world are nonlinear to some extent and either exhibit chaotic behavior or can be made to exhibit it.
		In fact, chaos is observed in so many systems in the real world that some scientists rank the understanding of chaos as being as important as the theories of relativity and quantum mechanics in that its ramifications stretch into every aspect of scientific study.
	dept.physics.upenn.edu /courses/gladney/mathphys/subsection3_2_5.html (1874 words)

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