Mathematical Biosciences Institute, Ohio State University, Columbus, Ohio 43210, USA.
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom.
Chaos. 2015 Sep;25(9):097612. doi: 10.1063/1.4918595.
We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts René Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network.
我们总结了过去三十年中关于对称动力系统和动力系统网络的一些主要研究成果，重点关注模式形成。在这两种情况下，都对动力系统施加了额外的约束，从而改变了一般现象。讨论的主要领域包括时变状态、模态相互作用以及非紧致对称群，如欧几里得群。我们考虑动力学和分岔。我们总结了这些思想在各种物理和生物系统中模式形成的应用，并解释了这些方法是如何通过将雷内·托姆的一般观点转移到新的背景中，其中一个版本被称为“突变理论”，来激发灵感的。我们强调了对称破缺在模式形成中的作用。主题包括等变 Hopf 分岔，它给出了从平衡态分岔出周期态的条件，以及 H/K 定理，它对等变动力学中周期态的集合对称和点对称对进行了分类。我们讨论了模态相互作用，它将多个分岔组织成一个单一的简并分岔，以及具有非紧致对称群的系统，其中会出现新的技术问题。我们将许多想法转移到耦合动力系统网络的背景中，并将网络动力学中的同步和相位关系解释为一种模式，其中空间离散化为有限数量的节点，而时间仍然是连续的。我们还描述了各种应用，包括动物运动、Couette-Taylor 流动、火焰、Belousov-Zhabotinskii 反应、双眼竞争以及基于前馈网络中周期性振荡幅度的异常增长率的非线性滤波器。
