Inaba Naohiko, Kousaka Takuji, Tsubone Tadashi, Okazaki Hideaki, Ito Hidetaka
Graduate School of Electrical and Information Engineering, Shonan Institute of Technology, Fujisawa 251-8511, Japan.
Department of Electrical and Electronic Engineering, Chukyo University, Nagoya 466-8666, Japan.
Chaos. 2021 Jul;31(7):073133. doi: 10.1063/5.0040658.
An extended Bonhoeffer-van der Pol (BVP) oscillator is a circuit that is naturally extended to a three-variable system from a two-variable BVP oscillator. A BVP oscillator is known to exhibit a canard explosion, and the extended BVP oscillator generates mixed-mode oscillations (MMOs). In this work, we considered a case study where the nonlinear conductor in the extended BVP oscillator includes an idealized diode. The idealized case corresponds to a degenerate case where one of the parameters tends to infinity, and circuit dynamics are represented using a constrained equation, and at the expense of the model's naturalness, i.e., in a case in which the solutions of the dynamics are defined only forward in time, the Poincaré return maps are constructed as one-dimensional (1D). Using these 1D return maps, we explain various phenomena, such as simple MMOs and MMO-incrementing bifurcations. In this oscillator, there exists a small amplitude oscillation, which emerges as a consequence of supercritical Hopf bifurcation, and there exists large relaxation oscillation which appears via canard explosion by changing the bifurcation parameter. Between these small and large amplitude oscillations, the MMO bifurcations exhibit asymmetric Farey trees. Furthermore, these theoretical results were verified using laboratory measurements and experiments.
扩展的邦霍夫尔 - 范德波尔（BVP）振荡器是一种从双变量BVP振荡器自然扩展到三变量系统的电路。已知BVP振荡器会出现鸭爆发现象，而扩展的BVP振荡器会产生混合模式振荡（MMO）。在这项工作中，我们考虑了一个案例研究，其中扩展BVP振荡器中的非线性导体包括一个理想化二极管。理想化情况对应于一种退化情况，即其中一个参数趋于无穷大，电路动力学用一个约束方程表示，并且以牺牲模型的自然性为代价，即在动力学解仅在时间上向前定义的情况下，庞加莱返回映射被构建为一维（1D）。使用这些一维返回映射，我们解释了各种现象，如简单MMO和MMO递增分岔。在这个振荡器中，存在由于超临界霍普夫分岔而出现的小振幅振荡，并且通过改变分岔参数，存在通过鸭爆发现象出现的大弛豫振荡。在这些小振幅和大振幅振荡之间，MMO分岔呈现出不对称的法雷树。此外，这些理论结果通过实验室测量和实验得到了验证。
