混沌系统相位同步的李雅普诺夫谱构造
Construction of the Lyapunov spectrum in a chaotic system displaying phase synchronization
摘要：
考虑了一个由Arnold猫映射的悬挂与钟通过弱耗散相互作用耦合而成的三维混沌系统。我们表明，耦合系统显示同步现象，在这个意义上，悬浮液流和时钟之间的相对相位锁定到一个特殊的值，从而使运动落在一个低维吸引子。更具体地说，我们构造的吸引力不变流形，尺寸小于3，使用收敛的微扰展开。此外，我们通过收敛级数计算的李雅普诺夫指数，特别是包括中心的。结果推广了以前的建设有吸引力的不变流形在一个类似的，但更简单的模型。目前建设的主要新颖性依赖于计算的李雅普诺夫谱，其中包括非平凡的分析指数。讨论了随着耦合度的增加，吸引子可能出现平滑过渡的问题。
英文摘要：
We consider a three-dimensional chaotic system consisting of the suspension of Arnold's cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.
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