Isaeva Olga B, Jalnine Alexey Yu, Kuznetsov Sergey P
Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelenaya 38, Saratov 410019, Russian Federation.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Oct;74(4 Pt 2):046207. doi: 10.1103/PhysRevE.74.046207. Epub 2006 Oct 9.
An example of a flow system is presented with an attractor concentrated mostly at a surface of a two-dimensional torus, the dynamics on which is governed by the Arnold cat map. The system is composed of four coupled nonautonomous van der Pol oscillators. Three of them have equal characteristic frequencies, and in the other one the frequency is twice as large. The parameters controlling excitation of the two pairs of oscillators are forced to undergo a slow counterphase periodic modulation in time. At the end of the active stage for one pair of the oscillators, the excitation is passed to another pair, than back, and so on. In terms of a stroboscopic Poincaré section, the respective eight-dimensional (8D) mapping, due to strong phase volume compression, reduces approximately to a 2D map for the phases of one pair of the oscillators that corresponds approximately to the Arnold cat map. The largest two Lyapunov exponents (one positive and one negative) are close to those predicted with the cat map model. Estimates for the fractal dimension of the attractor of the Poincaré map are close to 2.
给出了一个流动系统的示例，其吸引子主要集中在二维环面的一个表面上，该表面上的动力学由阿诺德猫映射控制。该系统由四个耦合的非自治范德波尔振荡器组成。其中三个具有相等的特征频率，另一个的频率是其两倍。控制两对振荡器激励的参数被迫随时间进行缓慢的反相周期调制。在一对振荡器的活跃阶段结束时，激励传递到另一对，然后再返回，如此循环。就频闪庞加莱截面而言，由于强烈的相体积压缩，相应的八维（8D）映射大约简化为一对振荡器相位的二维映射，该映射近似对应于阿诺德猫映射。最大的两个李雅普诺夫指数（一个为正，一个为负）接近猫映射模型预测的值。庞加莱映射吸引子的分形维数估计接近2。
