Abstract
One of the predominant themes of nonlinear dynamics during the past twenty years has been the characterization of the 'routes to chaos'. Within generic families of vector fields, there are ubiquitous patterns that describe how qualitative properties of vector fields can change with a varying parameter. Bifurcation theory classifies these patterns, and they form the substrate for determining the routes to chaos that one expects to see in physical systems. The presence of symmetry typically alters the patterns that one observes, and it has been well understood that symmetry can lead to the persistence of new types of dynamical behaviour and bifurcation. This paper gives another, more extreme, example of how symmetry can affect the routes to chaos. In the systems that the authors describe, there are persistent bifurcations that lead directly from a 'trivial' steady state to chaotic attractors of small amplitude. These bifurcations are 'supercritical' in the sense that the attractors emerge from the bifurcating equilibrium and remain confined to arbitrarily small neighbourhoods of the equilibrium for small values of the bifurcation parameter. Their analysis relies upon studying the global properties of a four-parameter family of four-dimensional vector fields which have invariant manifolds close to the unit sphere in their four-dimensional phase spaces. A bewildering variety of dynamical behaviour can be found in this family of vector fields, all of which represents possible post-bifurcation behaviour of the original problem. This analysis indicates that there are intrinsic limits to classifying bifurcations and characterize the routes to chaos through algebraic calculations.