Instant Chaos

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 elds, there are ubiquitous patterns that describe how qualitative properties of vector elds can change with a varying parameter. Bifurcation theory classiies 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 behavior and bifurcation. This paper gives another, more extreme, example of how symmetry can aaect the routes to chaos. In the systems that we describe below, 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 connned to arbitrarily small neighborhoods of the equilibrium for small values of the bifurcation parameter. Our analysis relies upon studying the global properties of a four parameter family of four dimensional vector elds which have invariant manifolds close to the unit sphere in their four dimensional phase spaces. A bewildering variety of dynamical behavior can be found in this family of vector elds, all of which represents possible post-bifurcation behavior of the original problem. This analysis indicates that there are intrinsic limits to our ability to classify bifurcations and characterize the routes to chaos through algebraic calculations.