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Chaos theory

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Chaos theory

A plot of Lorenz attractor for values r = 28, σ = 10, b = 8/3
A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions.

Chaos theory is the field of study in mathematics that studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect.[1] Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general.[2] This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.[3] In other words, the deterministic nature of these systems does not make them predictable.[4][5] This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:[6]

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

Chaotic behavior exists in many natural systems, such as weather and climate.[7][8] This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in several disciplines, including meteorology, sociology, physics, engineering, economics, biology, and philosophy.

Contents

• Introduction 1
• Chaotic dynamics 2
• Sensitivity to initial conditions 2.1
• Topological mixing 2.2
• Density of periodic orbits 2.3
• Strange attractors 2.4
• Minimum complexity of a chaotic system 2.5
• Jerk systems 2.6
• Spontaneous order 3
• History 4
• Distinguishing random from chaotic data 5
• Applications 6
• Computer science 6.1
• Biology 6.2
• Other areas 6.3
• References 8
• Scientific literature 9
• 9.1 Articles
• Textbooks 9.2
• Semitechnical and popular works 9.3

Introduction

Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: How much uncertainty we are willing to tolerate in the forecast, how accurately we are able to measure its current state, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the solar system, 50 million years. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears to be random.[9]

Chaotic dynamics

The map defined by x → 4 x (1 – x) and y → (x + y) mod 1 displays sensitivity to initial x positions. Here, two series of x and y values diverge markedly over time from a tiny initial difference. Note, however, that the y coordinate is effectively only defined modulo one, so the square region is actually depicting a cylinder, and the two points are closer than they look

In common usage, "chaos" means "a state of disorder".[10] However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition originally formulated by Robert L. Devaney says that, for a dynamical system to be classified as chaotic, it must have these properties:[11]

1. it must be sensitive to initial conditions
2. it must be topologically mixing
3. it must have dense periodic orbits

Sensitivity to initial conditions

Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points with significantly different future paths, or trajectories. Thus, an arbitrarily small change, or perturbation, of the current trajectory may lead to significantly different future behavior.

In some cases, the last two properties in the above have been shown to actually imply sensitivity to initial conditions,[12][13] and if attention is restricted to intervals, the second property implies the other two[14] (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list).[15] The most practically significant property, sensitivity to initial conditions, is redundant in the definition, since it is implied by two (or for intervals, one) purely topological properties, which are therefore of greater interest to mathematicians.

Sensitivity to initial conditions is popularly known as the "butterfly effect", so-called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C., entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

A consequence of sensitivity to initial conditions is that if we start with only a finite amount of information about the system (as is usually the case in practice), then beyond a certain time the system will no longer be predictable. This is most familiar in the case of weather, which is generally predictable only about a week ahead.[16] Of course, this does not mean that we cannot say anything about events far in the future; some restrictions on the system are present. With weather, we know that the temperature will never reach 100 °C or fall to -130 °C on earth, but we are not able to say exactly what day we will have the hottest temperature of the year.

In more mathematical terms, the Lyapunov exponent measures the sensitivity to initial conditions. Given two starting trajectories in the phase space that are infinitesimally close, with initial separation \delta \mathbf{Z}_0 end up diverging at a rate given by

| \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |\

where t is the time and λ is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.

Also, other properties relate to sensitivity of initial conditions, such as measure-theoretical mixing (as discussed in ergodic theory) and properties of a K-system.[5]

Topological mixing

The map defined by x → 4 x (1 – x) and y → (x + y) mod 1 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 vertical lines scattered across the space.

Topological mixing (or topological transitivity) means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 will tend to positive or negative infinity.

Density of periodic orbits

For a chaotic system to have a dense periodic orbit means that every point in the space is approached arbitrarily closely by periodic orbits.[17] The one-dimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits. For example, \tfrac{5-\sqrt{5}}{8} → \tfrac{5+\sqrt{5}}{8} → \tfrac{5-\sqrt{5}}{8} (or approximately 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).[18]

Sharkovskii's theorem is the basis of the Li and Yorke[19] (1975) proof that any one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits.

Strange attractors

The Lorenz attractor displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.

Some dynamical systems, like the one-dimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor, since then a large set of initial conditions will lead to orbits that converge to this chaotic region.

An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it was not only one of the first, but it is also one of the most complex and as such gives rise to a very interesting pattern, that with a little imagination, looks like the wings of a butterfly.

Unlike fixed-point attractors and limit cycles, the attractors that arise from chaotic systems, known as strange attractors, have great detail and complexity. Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points – Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and the fractal dimension can be calculated for them.

Minimum complexity of a chaotic system

Bifurcation diagram of the logistic map xr x (1 – x). Each vertical slice shows the attractor for a specific value of r. The diagram displays period-doubling as r increases, eventually producing chaos.

Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it has to be either nonlinear or infinite-dimensional.

The Poincaré–Bendixson theorem states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed above is generated by a system of three differential equations such as:

\begin{align} \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\ \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z. \end{align}

where x, y, and z make up the system state, t is time, and \sigma, \rho, \beta are the system parameters. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the Rossler equations which have only one nonlinear term out of seven. Sprott [20] found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel [21][22] showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved.

While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behavior.[23] Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.[24] A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.

Jerk systems

In physics, jerk is the third derivative of position, and as such, in mathematics differential equations of the form

J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0

are sometimes called Jerk equations. It has been shown, that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations is in a certain sense the minimal setting for solutions showing chaotic behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.[25]

A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits may be designed which model the solutions to this equation. These circuits are known as jerk circuits.

One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three first-order differential equations, but which may be combined into a single (although rather complicated) jerk equation. Nonlinear jerk systems are in a sense minimally complex systems to show chaotic behaviour, there is no chaotic system involving only two first-order, ordinary differential equations (the system resulting in an equation of second order only).

An example of a jerk equation with nonlinearity in the magnitude of x is:

\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}+A\frac{\mathrm{d}^2 x}{\mathrm{d} t^2}+\frac{\mathrm{d} x}{\mathrm{d} t}-|x|+1=0.

Here, A is an adjustable parameter. This equation has a chaotic solution for A=3/5 and can be implemented with the following jerk circuit; the required nonlinearity is brought about by the two diodes:

In the above circuit, all resistors are of equal value, except R_A=R/A=5R/3, and all capacitors are of equal size. The dominant frequency will be 1/2\pi R C. The output of op amp 0 will correspond to the x variable, the output of 1 will correspond to the first derivative of x and the output of 2 will correspond to the second derivative.

Spontaneous order

Under the right conditions, chaos will spontaneously evolve into a lockstep pattern. In the Kuramoto model, four conditions suffice to produce synchronization in a chaotic system. Examples include the coupled oscillation of Christiaan Huygens' pendulums, fireflies, neurons, the London Millenium Bridge resonance, and large arrays of Josephson junctions.[26]

History

Barnsley fern created using the chaos game. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an Iterated function system (IFS).

An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.[27][28] 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, called "Hadamard's billiards".[29] Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive Lyapunov exponent.

Chaos theory got its start in the field of

• Nonlinear Dynamics Research Group with Animations in Flash
• The Chaos group at the University of Maryland
• The Chaos Hypertextbook. An introductory primer on chaos and fractals
• Society for Chaos Theory in Psychology & Life Sciences
• Nonlinear Dynamics Research Group at CSDC, Florence Italy
• Interactive live chaotic pendulum experiment, allows users to interact and sample data from a real working damped driven chaotic pendulum
• Nonlinear dynamics: how science comprehends chaos, talk presented by Sunny Auyang, 1998.
• Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
• (excerpt)ChaosGleick's
• Systems Analysis, Modelling and Prediction Group at the University of Oxford
• A page about the Mackey-Glass equation
• High Anxieties — The Mathematics of Chaos (2008) BBC documentary directed by David Malone
• The chaos theory of evolution - article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos.
• Jos Leys, Étienne Ghys et Aurélien Alvarez, Chaos, A Mathematical Adventure. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.