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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク synchronization of chaos ： ウィキペディア英語版 synchronization of chaos Synchronization of chaos is a phenomenon that may occur when two, or more, dissipative chaotic systems are coupled. Because of the exponential divergence of the nearby trajectories of chaotic system, having two chaotic systems evolving in synchrony might appear surprising. However, synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and reasonably well understood theoretically. The stability of synchronization for coupled systems can be analyzed using Master Stability. Synchronization of chaos is a rich phenomenon and a multi-disciplinary discipline with broad range applications.〔A. Arenas, et al., Physics Rep. 469, 93 (2008).〕〔C. W. Wu, Synchronization in complex networks of nonlinear dynamical systems, World Scientific Publishing Co. Pte. Ltd., Singapore (2007)〕 Synchronization may present a variety of forms depending on the nature of the interacting systems and of the coupling scheme. == Identical synchronization == This type of synchronization is also known as complete synchronization. It can be observed for identical chaotic systems. The systems are said to be completely synchronized when there is a set of initial conditions so that the systems eventually evolve identically in time. In the simplest case of two diffusively coupled dynamics is described by ::$x^\; =\; F(x)\; +\; \backslash alpha(y-x)$ ::$y^\; =\; F(y)\; +\; \backslash alpha(x-y)$ where $F$ is the vector field modeling the isolated chaotic dynamics and $\backslash alpha$ is the coupling parameter. The regime $x(t)\; =\; y(t)$ defines an invariant subspace of the coupled system, if this subspace $x(t)\; =\; y(t)$ is locally attractive then the coupled system exhibit identical synchronization. If the coupling vanishes the oscillators are decoupled, and the chaotic behavior leads to a divergence of nearby trajectories. Complete synchronization occurs due to the interaction, if the coupling parameter is large enough so that the divergence of trajectories of interacting systems due to chaos is suppressed by the diffusive coupling. To find the critical coupling strength we study the behavior of the difference $v\; =\; x\; -\; y$. Assuming that $v$ is small we can expand the vector field in series and obtain a linear differential equation - by neglecting the taylor remainder - governing the behavior of the difference ::$v^\; =\; DF(x(t))\; v\; -\; 2\; \backslash alpha\; v$ where $DF(x(t))$ denotes the Jacobian of the vector field along the solution. If $\backslash alpha\; =\; 0$ then we obtain ::$u^\; =\; DF(x(t))\; u,$ and since the dynamics of chaotic we have $\backslash |\; u(t)\; \backslash |\; \backslash le\; \backslash |\; u(0)\; \backslash |\; e^$, where $\backslash lambda$ denotes the maximum Lyapunov exponent of the isolated system. Now using the ansatz $v\; =\; u\; e^$ we pass from the equation for $v$ to the equation for $u$. Therefore, we obtain ::$\backslash |v(t)\; \backslash |\; \backslash le\; \backslash |u(0)\backslash |\; e^$ yield a critical coupling strength $\backslash alpha\_c\; =\; \backslash lambda/2$, for all $\backslash alpha\; >\; \backslash alpha\_c$ the system exhibit complete synchronization. The existence of a critical coupling strength is related to the chaotic nature of the isolated dynamics. In general, this reasoning leads to the correct critical coupling value for synchronization. However, in some cases one might observe loss of synchronization for coupling strengths larger than the critical value. This occurs because the nonlinear terms neglected in the derivation of the critical coupling value can play an important role and destroy the exponential bound for the behavior of the difference.〔Peter Ashwin (2006), Scholarpedia, 1(8):1725 "()"〕 It is however, possible to give a rigorous treatment to this problem and obtain a critical value so that the nonlinearities will not affect the stability.〔Tiago Pereira, ''(Stability of Synchronized Motion in Complex Networks )'', arXiv:1112.2297v1, 2011.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「synchronization of chaos」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.