THE STRUCTURE OF INFINITE PERIODIC AND CHAOTIC HUB CASCADES IN PHASE DIAGRAMS OF SIMPLE AUTONOMOUS FLOWS

Abstract

This manuscript reports numerical investigations about the relative abundance and structure of chaotic phases in autonomous dissipative flows, i.e. in continuous-time dynamical systems described by sets of ordinary differential equations. In the first half, we consider flows containing "periodicity hubs", which are remarkable points responsible for organizing the dynamics regularly over wide parameter regions around them. We describe isolated hubs found in two forms of Rössler's equations and in Chua's circuit, as well as surprising infinite hub cascades that we found in a polynomial chemical flow with a cubic nonlinearity. Hub cascades converge orderly to accumulation points lying on specific parameter paths. In sharp contrast with familiar phenomena associated with unstable orbits, hubs and infinite hub cascades always involve stable periodic and chaotic orbits which are, therefore, directly measurable in experiments. In the last part, we consider flows having no hubs but unusual phase diagrams: a cubic polynomial model containing T-points and wide regions of dense chaos, a nonpolynomial model of the Belousov–Zhabotinsky reaction and the Hindmarsh–Rose model of neuronal bursting, both having chaotic phases with "fountains of chaos". The chaotic regions for the flows discussed here are different from what is known for discrete-time maps. This forcefully shows that knowledge about phase diagrams is quite fragmentary and that much work is still needed to classify and to understand them.