Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Abstract

We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of three-dimensional Mirá map.

On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators

“…Refs. [15][16][17][18][19][20][21] , however reasons for the appearance of such chaotic attractors in many cases are not clear. As far as we know, the corresponding explanation is given only in the following two cases: (i) when a system is subjected to an external quasiperiodic or skew forcing, see e.g.…”

“…Ref. 18 ; (ii) when a system has a lower dimension due to the presence of additional invariants, for example, first integral 22,23 (iii) when chaotic attractors exist in some neighborhood of a codimension-three bifurcation in the parameter space 17,20,21 .…”