Systems with Phase Space Dimension N ≥ 3: Deterministic Chaos

Abstract

The transition from the phase plane to a space of higher dimension leads to fundamental qualitative changes. The number of possible bifurcations of equilibrium states and limit cycles increases significantly, and many of them have not yet been studied. Some saddle sets become possible, such as an equilibrium state of the saddle-focus type and a saddle limit cycle. A cycle of the saddle-focus type and a saddle torus can be realized in a phase space with dimension N ≥ 4. The appearance of multi-dimensional stable and unstable manifolds of saddle sets and new types of doubly asymptotic trajectories such as separatrix loops of saddle foci and Poincaré homoclinic curves, leads in many cases to a complex structure in the phase portrait of a DS. The different kinds of behaviour actually realized are much more complex and varied. Besides periodic oscillations, quasiperiodic and chaotic oscillations can be observed. New types of attractors can emerge, namely, two-dimensional and multi-dimensional tori corresponding to quasiperiodic regimes, and strange chaotic attractors, which are the signature of dynamical chaos. Special types of DS behavior and special ‘exotic’ attractors can be observed under certain conditions, viz., strange nonchaotic and chaotic nonstrange attractors.