Resonances in heteroclinic networks

对称性（反射、旋转、平移）在理解许多物理系统的动力学和进化中起着重要作用。对称性的一个作用是允许不变量子空间的存在：例如，如果一个系统从反射对称状态开始，它将一直保持这种对称性（除非对称性被不完美或外部噪声破坏）。在理解具有对称性的物理系统在外力驱动下可能表现出的行为类型方面，已经取得了很大进展。最简单的行为类型，稳定和时间周期振荡，被很好地理解，并且这些状态之间的转换可以使用等变分岔理论进行分析。间断性还没有得到很好的理解。在这里，一个系统大部分时间都表现出一种行为，但偶尔会迅速转向另一种行为。间歇性可以通过结构稳定的异斜循环机制与对称性联系起来，其中动力系统平衡解之间的联系是鲁棒的，因为它们发生在对称不变的子空间中。在强健的异斜周期附近，动力学通常是间歇性的：系统在一个平衡点附近停留一段时间，迅速跳到第二个平衡点，并在那里停留更长的时间，在返回到原始平衡点之前继续循环，并在那里停留更长时间。如果循环是吸引的，在每个平衡点附近的停留时间呈几何级数无界限地增加，导致系统同时是高度间歇的，难以在数值上跟踪。在许多重要的例子中，异斜环是一个更大的异斜网络的一部分，其中两个或多个不同的环具有一些共同的平衡（或其他解）和连接，因此经过网络公共部分附近的轨道可以在网络中不同环的短途旅行之间切换。关于异斜环的稳定性，一些一般的结果是已知的，而一个环可以改变稳定性的更有趣的方式之一是通过共振分岔。然而，在异诊所网络的背景下，共振分岔知之甚少。重要的问题包括：当整个网络变得不稳定时会发生什么？是否有可能定义一个网络的共振，如果可以，与这种现象相关的动力学是什么？如果网络中只有一个周期有共振分叉，会有什么不同吗？这些都是有趣的问题，但人们对它们知之甚少。要想很好地理解异斜网络中的共振现象，主要障碍之一是，异斜网络中至少有一个元素必须具有大于1维数的不稳定流形，这使通常的分析方法变得非常复杂。我们建议通过数值和分析技术来研究各种异斜网络，旨在了解与共振分岔相关的动力学类型。我们还将研究轨迹如何在网络的不同部分之间切换，以及切换如何受到噪声和强制对称性破缺的影响。

Symmetries (reflections, rotations, translations) play a prominent role in understanding the dynamics and evolution of many physical systems. One effect of symmetry is to permit the existence of invariant subspaces: if a system is started in a reflection-symmetric state, for example, it will keep that symmetry for all time (unless the symmetry is broken by imperfections or external noise). Much progress has been made in understanding of types of behaviour a physical system with symmetry might exhibit as it is driven by external forcing. The simplest types of behaviour, steady and time-periodic oscillatory, are well understood, and transitions between such states can be analysed using equivariant bifurcation theory. Intermittency is less well understood. Here, a system spends most of its time exhibiting one type of behaviour, but has occasional rapid excursions to another type.Intermittency can be associated with symmetry through the mechanism of structurally stable heteroclinic cycles, in which connections between equilibrium solutions of a dynamical system are robust because they occur within a symmetry-invariant subspace. Near a robust heteroclinic cycle, the dynamics is typically intermittent: the system spends some time near one equilibrium point, jumps rapidly to a second equilibrium point and stays there for a longer time, continuing around the cycle before returning to the original equilibrium point and spending longer still there. If the cycle is attracting, the residence times close to each equilibrium point increase geometrically without bound, resulting in systems that are at once highly intermittent and difficult to follow numerically.In many important examples, heteroclinic cycles occur as part of a larger heteroclinic network, in which two or more distinct cycles have some common equilibria (or other solutions) and connections, so that orbits passing near the common parts of the network may switch between excursions around the different cycles in the network. Some general results are known about the stability of heteroclinic cycles, and one of the more interesting ways a cycle can change stability is through a resonant bifurcation.However, little is yet known about resonant bifurcations in the context of heteroclinic networks. Important questions include: What happens when a network as a whole becomes unstable? Is it possible to define resonance for a network, and if so what dynamics is associated with this phenomenon? Is something different seen if just one cycle within a network has a resonance bifurcation? These are all interesting questions about which little is understood.One of the main obstructions to a good understanding of resonance phenomena in the context of a heteroclinic network is that at least one element of the heteroclinic network must have an unstable manifold of dimension greater than one, and this greatly complicates the usual method of analysis. We propose to examine a variety of heteroclinic networks, both numerically and via analytical techniques, aiming to understand what kind of dynamics is associated with resonant bifurcations. We will also investigate how trajectories switch between different parts of the network, and how switching is influenced by noise and by forced symmetry breaking.