Structurally Stable Heteroclinic Cycles and the Dynamo Dynamics

Abstract

Heteroclinic cycles, i.e. trajectories that connect a finite number of saddle points of a dynamical system until they eventually come back to the same saddle point, are structurally unstable. They occur as bifurcation phenomena. However it has been shown that additional structure in the dynamical systems may lead to structurally stable behavior of these cycles. This is typically the case for Hamiltonian systems where it has been well known for a long time. In addition, symmetry in the equations will also force heteroclinic cycles to be structurally stable. This fundamentally is accomplished by the fact that symmetric systems will have invariant subspaces. Hence a connection between two saddles will become structurally stable if the restriction of one of the saddles to an invariant subspace leads to a sink in that subspace and hence the restriction of the flow to the invariant subspace may generate a saddle — sink connection. For a bibliography on the subject see [8] in the present volume and for a comprehensive introduction, see [6]. The prototypical example has been studied by Busse et al [4] and analysed as a robust heteroclinic cycle by Guckenheimer and Holmes [7]. Figure 1 illustrates the case: We consider a 3-d system where all coordinate planes and all coordinate axes are invariant subspaces. This can for instance be obtained for a system that has the symmetry group generated by reflections through the planes of coordinates and by cyclic permutation of these axes of coordinates. Now, assume that there exists a saddle on a coordinate axis with a 2-d stable manifold and a 1-d unstable manifold. By the permutation symmetry we will have saddles on each of the axes and if there is a heteroclinic orbit connecting 2 saddles in one invariant subspace there will be a whole cycle of these orbits connecting a saddle back to itself (see figure 1). In some parameter regimes the cycle is attracting. A time series for any of the three variables will show the variable to level off at a particular value until it transits to another value in a very short time where it will stay again for a long time etc. Upon addition of noise a“stochastic limit cycle” is created, whereby the transition times between saddles is exponentially distributed and unlike an attracting heteroclinic cycle without noise, there exists a finite mean period [11]. This dynamical behavior of relatively long quiescent behavior randomly followed by a quick transition to another long quiescent behavior makes this an attractive model for the behavior of magnetic reversals. The following sections will discuss our program to flesh out this model with more and more concrete physical details.