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演化中起着重要作用。对称性的一个效应是允许存在不变子空间：例如，如果一个系统在反射对称状态下启动，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受外力驱动时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，是很好的理解，和这种状态之间的转换可以使用等变分岔理论进行分析。间歇性不太好理解。间歇性可以通过结构稳定的异宿环机制与对称性联系在一起，在这种机制中，动力系统的平衡解之间的联系是稳健的，因为它们发生在一个不变量子空间内。在一个稳健的异宿周期附近，动力学通常是间歇性的：系统在一个平衡点附近花费一些时间，迅速跳到第二个平衡点并在那里停留更长时间，在返回到原始平衡点之前继续循环，并在那里花费更长时间。如果循环是吸引的，则接近每个平衡点的停留时间无限制地呈几何级数增加，导致系统同时具有高度间歇性，难以在数值上跟踪。在许多重要的例子中，异宿循环作为较大的异宿网络的一部分出现，其中两个或多个不同的循环具有一些共同的平衡点。（或其他解决方案）和连接，使得在网络的公共部分附近经过的轨道可以在围绕网络中的不同周期的偏移之间切换。关于异宿环的稳定性已经有了一些一般性的结果，而一个环改变稳定性的一个更有趣的方式是通过共振分岔，然而，对于异宿网络中的共振分岔还知之甚少。重要的问题包括：当整个网络变得不稳定时会发生什么？是否有可能为网络定义共振，如果有，什么动力学与这种现象有关？如果网络中只有一个循环有共振分叉，会有什么不同吗？这些都是有趣的问题，但人们对此知之甚少。在异宿网络中很好地理解共振现象的主要障碍之一是，异宿网络中至少有一个元素必须有一个维数大于1的不稳定流形，这使得通常的分析方法变得非常复杂。我们建议检查各种异宿网络，无论是通过数值和分析技术，旨在了解什么样的动力学与共振分岔。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.