arXiv 1504.019292015-04nlin.CD

耗散非自治系统中混沌边缘作为临界局域对称性破缺

Edge of chaos as critical local symmetry breaking in dissipative nonautonomous systems

Ricardo Chacón

AI总结：

本文提出几何共振解的局域对称性及其临界破缺可作为耗散非自治系统中混沌不稳定性发生的新解析判据，并解释了Melnikov方法在微扰区域之外依然有效的原因。

摘要：

本文在受空间周期相位调制势作用的耗散系统的一般背景下，讨论了完全非线性的共振概念——几何共振。研究表明，每个几何共振解都存在一个精确的局域不变量，当势场为静态时，该不变量退化为系统的能量。几何共振解代表一种局域对称性，其临界破缺为混沌不稳定性的发生提供了一个新的解析判据。这一物理判据是在共动参考系中，基于最短显著时间尺度上的局域能量守恒推导得出的。值得注意的是，这一关于混沌不稳定性发生的新物理判据被证明在参数空间的大范围内有效，因此其适用范围超越了当前数学技术所能覆盖的范围。更重要的是，本理论有助于理解Melnikov方法在微扰区域之外具有超常有效性的原因。

英文摘要：

The fully nonlinear notion of resonance$-$\textit{geometrical resonance}$-$in the general context of dissipative systems subjected to spatially periodic \textit{phase-modulated} potentials is discussed. It is demonstrated that there is an exact local invariant associated with each geometrical resonance solution which reduces to the system's energy when the potential is stationary. The geometrical resonance solutions represent a \textit{local symmetry} whose critical breaking leads to a new analytical criterion for the onset of chaotic instabilities. This physical criterion is deduced in the co-moving frame from the local energy conservation over the shortest significant timescale. Remarkably, the new physical criterion for the onset of chaotic instabilities is shown to be valid over large regions of parameter space, thus being useful beyond the scope of current mathematical techniques. More importantly, the present theory helps to understand the unreasonable effectiveness of the Melnikov's method beyond the perturbative regime.

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