Kirst Christoph, Timme Marc
Network Dynamics Group, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37073 Göttingen, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 2):065201. doi: 10.1103/PhysRevE.78.065201. Epub 2008 Dec 4.
We present a dynamical system that naturally exhibits two unstable attractors that are completely enclosed by each other's basin volume. This counterintuitive phenomenon occurs in networks of pulse-coupled oscillators with delayed interactions. We analytically show that upon continuously removing a local noninvertibility of the system, the two unstable attractors become a set of two nonattracting saddle states that are heteroclinically connected. This transition equally occurs from larger networks of unstable attractors to heteroclinic structures and constitutes a new type of singular bifurcation in dynamical systems.
我们展示了一个动力学系统，该系统自然地呈现出两个不稳定吸引子，它们完全被彼此的盆地体积所包围。这种违反直觉的现象出现在具有延迟相互作用的脉冲耦合振荡器网络中。我们通过分析表明，在持续消除系统的局部不可逆性时，这两个不稳定吸引子会变成一组通过异宿连接的非吸引鞍点状态。这种转变同样会从更大的不稳定吸引子网络发生到异宿结构，并且构成了动力学系统中的一种新型奇异分岔。
