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Что (кто) такое chaos theory - определение
![Soviet physicist]] [[Lev Landau]], who developed the [[Landau-Hopf theory of turbulence]]. [[David Ruelle]] and [[Floris Takens]] later predicted, against Landau, that [[fluid turbulence]] could develop through a [[strange attractor]], a main concept of chaos theory.](https://commons.wikimedia.org/wiki/Special:FilePath/Airplane vortex edit.jpg?width=200 "Soviet physicist]] [[Lev Landau]], who developed the [[Landau-Hopf theory of turbulence]]. [[David Ruelle]] and [[Floris Takens]] later predicted, against Landau, that [[fluid turbulence]] could develop through a [[strange attractor]], a main concept of chaos theory.")
![[[Barnsley fern]] created using the [[chaos game]]. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an [[iterated function system]] (IFS).](https://commons.wikimedia.org/wiki/Special:FilePath/Barnsley fern plotted with VisSim.PNG?width=200 "[[Barnsley fern]] created using the [[chaos game]]. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an [[iterated function system]] (IFS).")
![mod]] 1</span> displays sensitivity to initial x positions. Here, two series of ''x'' and ''y'' values diverge markedly over time from a tiny initial difference.](https://commons.wikimedia.org/wiki/Special:FilePath/Chaos Sensitive Dependence.svg?width=200 "mod]] 1</span> displays sensitivity to initial x positions. Here, two series of ''x'' and ''y'' values diverge markedly over time from a tiny initial difference.")
![mod]] 1</span> 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.](https://commons.wikimedia.org/wiki/Special:FilePath/Chaos Topological Mixing.png?width=200 "mod]] 1</span> 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.")
![double-rod pendulum]] at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different [[initial condition]] would result in a vastly different [[trajectory]]. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.](https://commons.wikimedia.org/wiki/Special:FilePath/Double-compound-pendulum.gif?width=200 "double-rod pendulum]] at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different [[initial condition]] would result in a vastly different [[trajectory]]. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.")
![Six iterations of a set of states <math>[x,y]</math> passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that ''mixing'' occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation <math>x_{k+1} = 4 x_k (1 - x_k )</math>. To expand the state-space of the logistic map into two dimensions, a second state, <math>y</math>, was created as <math>y_{k+1} = x_k + y_k </math>, if <math>x_k + y_k <1</math> and <math>y_{k+1} = x_k + y_k -1</math> otherwise.](https://commons.wikimedia.org/wiki/Special:FilePath/LogisticTopMixing1-6.gif?width=200 "Six iterations of a set of states <math>[x,y]</math> passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that ''mixing'' occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation <math>x_{k+1} = 4 x_k (1 - x_k )</math>. To expand the state-space of the logistic map into two dimensions, a second state, <math>y</math>, was created as <math>y_{k+1} = x_k + y_k </math>, if <math>x_k + y_k <1</math> and <math>y_{k+1} = x_k + y_k -1</math> otherwise.")
![period-doubling]] as ''r'' increases, eventually producing chaos. Darker points are visited more frequently.](https://commons.wikimedia.org/wiki/Special:FilePath/Logistic Map Bifurcation Diagram, Matplotlib.svg?width=200 "period-doubling]] as ''r'' increases, eventually producing chaos. Darker points are visited more frequently.")
![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.](https://commons.wikimedia.org/wiki/Special:FilePath/TwoLorenzOrbits.jpg?width=200 "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.")
Википедия
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.
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. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. 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. It also occurs spontaneously in some systems with artificial components, such as the road traffic. This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology, anthropology, sociology, environmental science, computer science, engineering, economics, ecology, and pandemic crisis management. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes.
