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陈振阳, 韩修静, 毕勤胜. 离散达芬映射中由边界激变所诱发的复杂的张弛振荡[J]. 力学学报, 2017, 49(6): 1380-1389. DOI: 10.6052/0459-1879-17-138
引用本文: 陈振阳, 韩修静, 毕勤胜. 离散达芬映射中由边界激变所诱发的复杂的张弛振荡[J]. 力学学报, 2017, 49(6): 1380-1389. DOI: 10.6052/0459-1879-17-138
Chen Zhenyang, Han Xiujing, Bi Qinsheng. COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(6): 1380-1389. DOI: 10.6052/0459-1879-17-138
Citation: Chen Zhenyang, Han Xiujing, Bi Qinsheng. COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(6): 1380-1389. DOI: 10.6052/0459-1879-17-138

离散达芬映射中由边界激变所诱发的复杂的张弛振荡

COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP

  • 摘要: 多时间尺度问题具有广泛的工程与科学研究背景,慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题,其展现出的振荡形式及内部分岔结构,大多较为单一,此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例,探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线,其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下,存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时,混沌吸引子可能与不稳定不动点相接触,也可能与之相距一定距离.对快子系统吸引域分布的模拟,表明存在着导致边界激变(boundary crisis)的临界值,在这些值附近,经由延迟倍周期分岔演化而来的混沌吸引子可与2nn=0,1,2,…)周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后,双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁,从而使系统出现了不同吸引子之间的滞后行为,由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.

     

    Abstract: Multiple-time scale problems are ubiquitous in both science and engineering, while the slow varying parameter is one of the iconic feature of multiple-time scale. However, up till now, most of bifurcation structures and oscillation patterns revealed by literatures are relatively simplex. In this paper, we take the non-autonomous Duffing map as a example to explore family of complex relaxation oscillation patterns, which are little concerned by previous study. The fast subsystem exhibits an S-shaped fixed point curve, and the stable upper and lower branches evolve into chaos by a cascade of Flip bifurcations. What's more, we can observe a pair of critical parameter values under some parameter conditions, which lead to the catastrophe vanish of chaotic attractors. When the bifurcation parameter reaches these values, chaotic attractors may contact with the unstable fixed point or just stay in a distance apart. By simulating the distribution of basins of attraction owned by fast subsystem, we show that there exist critical points of boundary crisis, nearby which chaotic attractor evolved from stable fixed points can coexist with period-2n (n=0, 1, 2, …) attractor or even another chaotic attractor. When the non-autonomous term (i.e., the slow variable) passes through critical points, distruction of bi-stability may lead to the transition from chaotic attractor in pre-crisis stage to the coexisting attractor, thus the boundary crisis motivates different patterns of symmetric relaxation oscillation. In particular, patterns here show structures containing different number of delay flip bifurcations, owe to the fact that delay quantities of Flip points in different level take disparate magnitude. Our results enrich dynamical mechanisms of multiple-time scale in discrete systems.

     

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