Riddled Basins and Coupled Dynamical Systems

Abstract

The collective dynamics of groups of coupled dynamical systems is of great interest for understanding spontaneous pattern formation in biological and many other systems; see for example [58]. One can learn a lot about such systems by first studying idealized cases where the systems are perfectly identical; this approach has been very successful in understanding general properties of synchronization as well as particular applications; see for example [77]. In this chapter we consider how this can lead to the appearance of attractors with riddled basins. These basins appear because symmetries of dynamical systems force the presence of invariant submanifolds; the attractors within invariant manifolds may be only weakly attracting transverse to the invariant manifold and this leads to a basin structure that is, roughly speaking, full of holes. From a theoretical point of view, this behaviour is of interest because it seems strange or pathological but is in some sense common. From a practical point of view, this behaviour points towards the presence of extreme sensitivity of the dynamics to noise, also called ‘bubbling’ of attractors. Most interestingly, if we consider generic dynamics within a class of symmetric systems, riddled basins can appear as a robust phenomenon; they can be persistent for open sets of parameters of the system. For the remainder of this section we briefly discuss basins of attraction and a motivating example of a piecewise linear map with an explicitly computable riddled basin attractor. More general properties of riddled sets and basins are discussed in Sect. 2 including their noise sensitivity. This is followed in Sect. 3 by a discussion of the use of symmetries, ergodic measures and Lyapunov exponents tools for identifying riddled basins; we also discuss anisotropic riddling in Sect. 4 along the lines of [7]. Finally in Sect. 5 we outline a few open problems related to riddling phenomena.