Isaeva Olga B, Kuznetsov Sergey P, Mosekilde Erik
Kotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Science, Saratov, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jul;84(1 Pt 2):016228. doi: 10.1103/PhysRevE.84.016228. Epub 2011 Jul 29.
The paper proposes an approach to constructing feasible examples of dynamical systems with hyperbolic chaotic attractors based on the successive transfer of excitation between two pairs of self-oscillators that are alternately active. An angular variable that measures the relations of the current amplitudes for the two oscillators of each pair undergoes a transformation in accordance with the expanding circle map during each cycle of the process. We start with equations describing the dynamics in terms of complex or real amplitudes and then examine two models based on van der Pol oscillators. One model corresponds to the situation of equality of natural frequencies of the partial oscillators, and another to a nonresonant ratio of the oscillation frequencies relating to each of the two pairs. Dynamics of all models are illustrated with diagrams indicating the transformation of the angular variables, portraits of attractors, Lyapunov exponents, etc. The uniformly hyperbolic nature of the attractor in the stroboscopic Poincaré map is confirmed for a real-amplitude version of the equations by computations of statistical distribution of angles between stable and unstable manifolds at a representative set of points on the attractor. In other versions of the equations the attractors relate presumably to the partially hyperbolic class.
本文提出了一种基于两对交替活跃的自激振荡器之间激励的连续传递来构建具有双曲混沌吸引子的动力系统可行示例的方法。在该过程的每个周期中，用于测量每对两个振荡器当前振幅关系的角变量会根据扩展圆映射进行变换。我们从用复振幅或实振幅描述动力学的方程开始，然后研究基于范德波尔振荡器的两个模型。一个模型对应于部分振荡器固有频率相等的情况，另一个对应于与两对中的每一对相关的振荡频率的非共振比。所有模型的动力学都通过图表进行说明，这些图表展示了角变量的变换、吸引子的图像、李雅普诺夫指数等。通过计算吸引子上一组代表性点处稳定和不稳定流形之间角度的统计分布，证实了频闪庞加莱映射中吸引子的一致双曲性质，该计算针对方程的实振幅版本进行。在方程的其他版本中，吸引子大概属于部分双曲类。
