Scenarios for the Creation of Hyperchaotic Attractors of 3D Maps

TL;DR

This paper explores bifurcation mechanisms leading to hyperchaotic attractors in 3D maps, proposing scenarios involving cascades of bifurcations, and illustrates them using the Mirá map example.

Contribution

It introduces new bifurcation scenarios for hyperchaos in 3D maps, emphasizing cascades of period-doubling and Neimark-Sacker bifurcations.

Findings

Identified bifurcation cascades leading to hyperchaos.

Demonstrated scenarios with the Mirá map.

Showed the role of two-dimensional unstable manifolds.

Abstract

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e., such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In order to possess this property periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. For realization of this possibility, we propose several bifurcation scenarios 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 the three-dimensional Mir\'a map.