A minimal system with a depth-two heteroclinic network
A minimal system with a depth-two heteroclinic network
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DOI：
10.1080/14689367.2010.500102
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发表时间：
2010-08
期刊：
影响因子：
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通讯作者：
Tsuyoshi Chawanya;P. Ashwin
中科院分区：
文献类型：
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作者：
Tsuyoshi Chawanya;P. Ashwin
我们给出了一个三维常微分方程解的例子，它的不变集是双曲平衡点之间的深度为2的异宿网络(具有层次结构的异宿网络)。这是第一个最小维度(即，三个维度)的示例，它对于受约束的ODE系统是健壮的-约束是流保持不变立方体的边界和通过立方体的相反面的轴。我们从解析和数值两个角度研究了系统，并在网络的一个邻域中得到了一个简化的一维返回映射，在那里我们观察到了各种复杂的动力学行为。这包括无穷多个稳定极限环的共存，我们提出了在这个邻域内可能有无限多个混沌吸引子的证据。
We present an example of a three-dimensional ordinary differential equation with an invariant set that is a heteroclinic network of depth two (a heteroclinic network with a hierarchical structure) between hyperbolic equilibria. This is the first example of minimal dimension (namely, three dimensions) that is robust for a constrained system of ODEs – the constraint is that the flow preserves the boundaries of an invariant cube and an axis through opposite faces of the cube. We examine the system from both analytical and numerical viewpoints, and in a neighbourhood of the network we derive a reduced, one-dimensional, return map where we observe a variety of complicated dynamical behaviours. This includes co-existence of infinitely many stable limit cycles, and we present evidence that there may be infinitely many chaotic attractors within this neighbourhood.