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Chaos theory is a branch of mathematics focusing on the study of chaos — dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.
The theory was summarized by Edward Lorenz as: . Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate.
Chaos theory has applications in a variety of disciplines, including meteorology ,  anthropology ,  sociology , physics ,  environmental science , computer science , engineering , economics , biology , ecology , pandemic crisis management ,   and philosophy. The theory formed the basis for such fields of study as complex dynamical systems , edge of chaos theory, and self-assembly processes.
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 that the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, 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 inner solar system, 4 to 5 million years. Hence, mathematically, 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 random. In common usage, "chaos" means "a state of disorder".
Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by Robert L. Devaney , says that to classify a dynamical system as chaotic, it must have these properties: . In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. If attention is restricted to intervals , the second property implies the other two.
In continuous time dynamical systems, chaos is the phenomenon of the spontaneous breakdown of topological supersymmetry, which is an intrinsic property of evolution operators of all stochastic and deterministic partial differential equations. Within this picture, the long-range dynamical behavior associated with chaotic dynamics e. Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points that have significantly different future paths or trajectories.
Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior. 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 to the American Association for the Advancement of Science in Washington, D.
Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different. A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system as is usually the case in practice , then beyond a certain time, the system would no longer be predictable.
This is most prevalent in the case of weather, which is generally predictable only about a week ahead. In more mathematical terms, the Lyapunov exponent measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions.
The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can 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. In addition to the above property, other properties related to sensitivity of initial conditions also exist.
These include, for example, measure-theoretical mixing as discussed in ergodic theory and properties of a K-system. A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact, will diverge from it.
Thus for almost all initial conditions, the variable evolves chaotically with non-periodic behavior. Topological mixing or the weaker condition of topological transitivity means that the system evolves over time so that any given region or open set of its phase space eventually overlaps 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 eventually becomes widely separated.
However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity. Topological transitivity is a weaker version of topological mixing. Intuitively, if a map is topologically transitive then given a point x and a region V , there exists a point y near x whose orbit passes through V.
This implies that is impossible to decompose the system into two open sets. An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies in topological transitivity. The Birkhoff Transitivity Theorem states that if X is a second countable , complete metric space , then topological transitivity implies the existence of a dense set of points in X that have dense orbits. For a chaotic system to have dense periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.
Sharkovskii's theorem is the basis of the Li and Yorke  proof that any continuous 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. The cases of most interest arise when the chaotic behavior takes place on an attractor , since then a large set of initial conditions leads 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 is 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. 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.
Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it must be either nonlinear or infinite-dimensional.
The Lorenz attractor discussed below is generated by a system of three differential equations such as:. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms.
Sprott  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   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. In physics , jerk is the third derivative of position , with respect to time.
As such, differential equations of the form. 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. A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits.
One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. 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. Here, A is an adjustable parameter.
The output of op amp 0 will correspond to the x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative. Similar circuits only require one diode  or no diodes at all. See also the well-known Chua's circuit , one basis for chaotic true random number generators. Under the right conditions, chaos spontaneously evolves 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 Millennium Bridge resonance, and large arrays of Josephson junctions. In the s, 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.
Chaos theory began in the field of ergodic theory. Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory , the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple " noise " was considered by chaos theorists as a full component of the studied systems.
The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems.
As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, , 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 Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course.
He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To his surprise, the weather the machine began to predict was completely different from the previous calculation.
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