Ashwin Peter, Field Michael, Rucklidge Alastair M, Sturman Rob
School of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, United Kingdom.
Chaos. 2003 Sep;13(3):973-81. doi: 10.1063/1.1586531.
In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated, owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena, including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. In this paper we introduce and discuss an instructive example of an ordinary differential equation where one can observe and analyze robust cycling behavior. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a Rössler system), and/or saddle equilibria. For this model, we distinguish between cycling that includes phase resetting connections (where there is only one connecting trajectory) and more general non(phase) resetting cases, where there may be an infinite number (even a continuum) of connections. In the nonresetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability, whereas more general cases may give rise to "stuck on" cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase resetting and connection selection.
在存在对称性或不变子空间的情况下，由于与不变子空间的相互作用，动力系统中的吸引子可能会变得非常复杂。这引发了许多新现象，包括呈现混沌巡游的鲁棒吸引子现象。在最简单的层面上，这是平衡点之间的一个吸引异宿环，但更一般的不变集之间的环也是可能的。在本文中，我们引入并讨论一个常微分方程的有启发性的例子，在其中可以观察和分析鲁棒的循环行为。通过设计，我们可以表明在不变集之间存在一个鲁棒环，这些不变集可能是混沌鞍点（其内部动力学对应于一个罗斯勒系统）和/或鞍点平衡点。对于这个模型，我们区分包含相位重置连接的循环（只有一条连接轨迹的情况）和更一般的非（相位）重置情况，在非重置情况中可能存在无限多个（甚至是连续统）连接。在非重置情况下存在连接选择的问题：对于典型的吸引轨迹会观察到哪些连接？我们讨论这种循环对李雅普诺夫指数共振的不稳定性，并将其与一个猜想相关联，即相位重置环在不稳定时通常会导致稳定的周期轨道，而更一般的情况可能会导致“被困”循环。最后，我们讨论混沌鞍点的正李雅普诺夫指数的存在意味着在解释返回时间变长的数值模拟时我们需要非常小心；这会严重影响相位重置和连接选择的模拟。
