强收缩系统的多模态振动
Multimodal oscillations in systems with strong contraction
摘要：
本文研究了$R^3$中Andronov-Hopf分岔（AHB）附近的单参数和双参数流动族。我们确定的全球向量场，这产生了丰富的家庭的多模态轨道通过接近一个弱不稳定的鞍焦点和鞍焦点附近的轨迹进行详细的渐近分析的条件。我们的分析涵盖了亚和超临界AHB的情况。对于超临界情形，我们发现，从AHB诞生的周期轨道是双峰时，在所产生的坐标系的线性化的分叉平衡。如果AHB是亚临界的，它是伴随着多模态轨道的外观，其中包括一长串的近谐波振荡分离的大幅度尖峰。我们分析了控制参数上的峰间期（可以是非常长的）的依赖性。特别是，我们表明，interspike间隔增长的亚和超临界AHB区域之间的边界接近的参数空间。我们还确定了一个窗口的子和超临界AHB区域之间的边界附近的复杂和可能的混沌振荡，并解释产生这些振荡的机制。这项工作的动机是一个有限维近似的自由边界问题模拟固体燃料燃烧的数值结果。
英文摘要：
One- and two-parameter families of flows in $R^3$ near an Andronov-Hopf bifurcation (AHB) are investigated in this work. We identify conditions on the global vector field, which yield a rich family of multimodal orbits passing close to a weakly unstable saddle-focus and perform a detailed asymptotic analysis of the trajectories in the vicinity of the saddle-focus. Our analysis covers both cases of sub- and supercritical AHB. For the supercritical case, we find that the periodic orbits born from the AHB are bimodal when viewed in the frame of coordinates generated by the linearization about the bifurcating equilibrium. If the AHB is subcritical, it is accompanied by the appearance of multimodal orbits, which consist of long series of nearly harmonic oscillations separated by large amplitude spikes. We analyze the dependence of the interspike intervals (which can be extremely long) on the control parameters. In particular, we show that the interspike intervals grow logarithmically as the boundary between regions of sub- and supercritical AHB is approached in the parameter space. We also identify a window of complex and possibly chaotic oscillations near the boundary between the regions of sub- and supercritical AHB and explain the mechanism generating these oscillations. This work is motivated by the numerical results for a finite-dimensional approximation of a free boundary problem modeling solid fuel combustion.
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