Pathways to hyperchaos in a three-dimensional quadratic map

TL;DR

This paper explores the mechanisms and routes leading to hyperchaos in a three-dimensional quadratic map, highlighting the transition from stable states to hyperchaotic regimes with three positive Lyapunov exponents.

Contribution

It identifies and analyzes multiple routes to hyperchaos, including bifurcations from fixed points and quasiperiodic curves, and examines the role of saddle periodic orbits in this process.

Findings

Hyperchaotic regimes exhibit three positive Lyapunov exponents over wide parameter ranges.

Saddle periodic orbits become repellers within hyperchaotic attractors.

Routes to hyperchaos include period-doubling and bifurcations of invariant curves.

Abstract

This paper deals with various routes to hyperchaos with all three positive Lyapunov exponents in a three-dimensional quadratic map. The map under consideration displays strong hyperchaoticity in the sense that in a wider range of parameter space the system showcase three positive Lyapunov exponents. It is shown that the saddle periodic orbits eventually become repellers at this hyperchaotic regime. By computing the distance of the repellers to the attractors as a function of parameters, it is shown that the hyperchaotic attractors absorb the repelling periodic orbits. First we discuss a route from stable fixed point undergoing period-doubling bifurcations to chaos and then hyperchaos, and role of saddle periodic orbits. We then illustrate a route from doubling bifurcation of quasiperiodic closed invariant curves to hyperchaotic attractors. Finally, presence of weak hyperchaotic flow like attractors are discussed.