Department of Chemistry, Brandeis University, Waltham, Massachusetts 02453, USA.
Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215, USA.
Chaos. 2023 Jan;33(1):011102. doi: 10.1063/5.0131305.
Symmetry-breaking in coupled, identical, fast-slow systems produces a rich, dramatic variety of dynamical behavior-such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast-slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel-Epstein model of chemical oscillators.
在耦合的、相同的快-慢系统中，对称破缺会产生丰富多样的动力学行为，例如幅度和频率相差一个数量级或更多，以及振荡器之间的节奏在性质上不同，对应于不同的功能状态。我们提出了一种分析这些系统的新方法。它确定了导致这种新的对称破缺的关键几何结构，并表明许多不同类型的对称破缺节奏会稳健地出现。我们发现了对称破缺节奏，其中一个振荡器表现出小幅度的振荡，而另一个振荡器则表现出相移的小幅度振荡、大幅度的振荡、混合模式的振荡，甚至经历极限环卡丹的爆炸。两个典型的快-慢系统说明了这种方法：描述电路的范德波尔方程和化学振荡器的伦格-爱泼斯坦模型。
