Abstract

This article focuses on the dynamics of a modified van der Pol–Duffing circuit (MVDPD hereafter) (Fotsin and Woafo in Chaos Solitons and Fractals 24(5):1363–1371, 2005) whose symmetry is explicitly broken with the presence an offset term. When ignoring offset terms, the system displays an exact symmetry which is reflected in the location of the equilibrium points, the attractor topologies and the attraction basins shapes as well. In this mode of operation, the system displays typical behaviors such as period doubling sequences; spontaneous symmetry breaking, symmetry recovering, and multistability involving several pairs of mutually symmetric attractors. In the presence of offset terms, the MVDPD circuit is non-symmetric and more complex nonlinear phenomena arise such as parallel bifurcation branches, coexisting multiple (i.e. two, three, four or five) asymmetric attractors, and crises. It should be noted that for each case of multistability discussed in this work, a hidden attractor (period-1 limit cycle) coexists with self-excited others. To the best of our knowledge, the coexistence of five attractors (symmetrical or asymmetrical), one of which is hidden has not yet been reported in the MVDPD circuit and thus deserves dissemination. PSpice simulation investigations based on the implementation of the MVDPD confirm the theoretical predictions.

中文翻译：

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改进的范德波尔-达芬振荡器的动力学，控制和对称性破坏方面及其模拟电路实现

摘要

本文关注于改进的范德波尔-达芬电路（以下简称MVDPD）的动力学（Fotsin和Woafo，见《混沌孤子和分形》 24（5）：1363–1371，2005），其对称性被存在偏移项的情况明确破坏了。当忽略偏移项时，系统显示精确的对称性，该对称性反映在平衡点的位置，吸引子拓扑结构和吸引盆的形状上。在这种操作模式下，系统显示典型的行为，例如周期倍增序列；自发对称破坏，对称恢复以及涉及几对相互对称吸引子的多重稳定性。在存在偏移项的情况下，MVDPD电路是非对称的，并且会出现更复杂的非线性现象，例如平行的分支分支，多个并存（即两个，三个，四或五）不对称的吸引子和危机。应当指出的是，对于本工作中讨论的每种多稳定性情况，一个隐藏的吸引子（周期为1的极限循环）与自激他人共存。据我们所知，MVDPD电路中尚未报告五个吸引子（对称或不对称）的共存，其中之一是隐藏的，因此值得传播。基于MVDPD实施的PSpice仿真研究证实了理论预测。MVDPD电路中尚未报告其中之一是隐藏的，因此值得传播。基于MVDPD实施的PSpice仿真研究证实了理论预测。MVDPD电路中尚未报告其中之一是隐藏的，因此值得传播。基于MVDPD实施的PSpice仿真研究证实了理论预测。