Maistrenko Y L, Maistrenko V L, Popovych O, Mosekilde E
Institute of Mathematics, National Academy of Sciences, Kiev, 252601, Ukraine.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1999 Sep;60(3):2817-30. doi: 10.1103/physreve.60.2817.
When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.
当相同的混沌振荡器相互作用时，可能会达到完全或部分同步的状态，其中运动被限制在一个维度低于全相空间的不变流形上。当嵌入同步混沌状态的特定轨道变得横向不稳定而该状态平均上仍保持吸引时，吸引盆就会出现“ riddling”现象。考虑一个由两个耦合逻辑斯谛映射组成的系统，我们表明，向“ riddling”的转变将是柔和的还是剧烈的，这取决于第一个失去横向稳定性的轨道经历的是超临界分岔还是亚临界分岔。亚临界分岔可直接导致同步混沌状态吸引盆的全局“ riddling”。另一方面，超临界分岔与沿着同步混沌状态延伸且轨迹无法从中逃脱的所谓混合吸收区域的形成有关。这导致了局部“ riddled”吸引盆。我们提出了三种不同的“ riddling”起始情况以及吸引盆随后的转变情况。每种情况通过跟踪相关异步周期的类型和位置，并确定它们的稳定和不稳定不变流形来描述。一种情况涉及吸引盆边界与吸收区域之间的接触分岔。另一种情况涉及一系列漫长而有趣的分岔，从“ riddling”分岔中产生的异步周期的稳定开始，到异步混沌状态的稳定性被破坏的边界危机结束。最后，给出了一个相图来说明各种转变发生时的参数值。
