Nonlinear oscillatory mixing in the generalized Landau scenario

Abstract

We present a set of phase-space portraits illustrating the extraordinary oscillatory possibilities of the dynamical systems through the so-called generalized Landau scenario. In its simplest form the scenario develops in N dimensions around a saddle-node pair of fixed points experiencing successive Hopf bifurcations up to exhausting their stable manifolds and generating N-1 different limit cycles. The oscillation modes associated with these cycles extend over a wide phase-space region by mixing ones within the others and by affecting both the transient trajectories and the periodic orbits themselves. A mathematical theory covering the mode-mixing mechanisms is lacking, and our aim is to provide an overview of their main qualitative features in order to stimulate research on it.

24 pages, 10 figures and 1 table. Submitted to Physica D and then to Physical Review E. Minor corrections added

References in corpus (2)

• Mixed-mode oscillations in a multiple time scale phantom bursting system
• Unphysical source of dynamic order for the natural world

Cited by in corpus (2)

• Chaos on a High-Dimensional Torus
• Designable Dynamical Systems for the Generalized Landau Scenario and the Nonlinear Complexification of Periodic Orbits