Deshpande Amogh, Chen Qingfei, Wang Yan, Lai Ying-Cheng, Do Younghae
School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Aug;82(2 Pt 2):026209. doi: 10.1103/PhysRevE.82.026209. Epub 2010 Aug 24.
In piecewise-smooth dynamical systems, situations can arise where the asymptotic attractors of the system in an open parameter interval are all chaotic (e.g., no periodic windows). This is the phenomenon of robust chaos. Previous works have established that robust chaos can occur through the mechanism of border-collision bifurcation, where border is the phase-space region where discontinuities in the derivatives of the dynamical equations occur. We investigate the effect of smoothing on robust chaos and find that periodic windows can arise when a small amount of smoothness is present. We introduce a parameter of smoothing and find that the measure of the periodic windows in the parameter space scales linearly with the parameter, regardless of the details of the smoothing function. Numerical support and a heuristic theory are provided to establish the scaling relation. Experimental evidence of periodic windows in a supposedly piecewise linear dynamical system, which has been implemented as an electronic circuit, is also provided.
在分段光滑动力系统中，可能会出现这样的情况：在一个开放参数区间内，系统的渐近吸引子全是混沌的（例如，没有周期窗口）。这就是鲁棒混沌现象。先前的研究已经表明，鲁棒混沌可以通过边界碰撞分岔机制发生，其中边界是动力方程导数出现不连续的相空间区域。我们研究了平滑对鲁棒混沌的影响，发现当存在少量平滑时会出现周期窗口。我们引入一个平滑参数，发现参数空间中周期窗口的度量与该参数呈线性比例关系，而与平滑函数的细节无关。提供了数值支持和一个启发式理论来建立这种比例关系。还给出了一个已实现为电子电路的假定分段线性动力系统中周期窗口的实验证据。
