Is a hyperchaotic attractor superposition of two multifractals?

TL;DR

This paper proposes that hyperchaotic attractors can be understood as dual multifractals, revealing a complex geometric structure that transitions from simple to dual multifractal during hyperchaos onset.

Contribution

It introduces the novel idea that hyperchaotic attractors are characterized by a superposition of two multifractal spectra, supported by numerical evidence and a new generation scheme.

Findings

Hyperchaotic attractors exhibit dual multifractal spectra.

Transition to hyperchaos involves a structural change to dual multifractality.

A method to generate hyperchaotic attractors by coupling low-dimensional systems.

Abstract

In the context of chaotic dynamical systems with exponential divergence of nearby trajectories in phase space, hyperchaos is defined as a state where there is divergence or stretching in at least two directions during the evolution of the system. Hence the detection and characterization of a hyperchaotic attractor is usually done using the spectrum of Lyapunov Exponents (LEs) that measure this rate of divergence along each direction. Though hyperchaos arise in different dynamical situations and find several practical applications, a proper understanding of the geometric structure of a hyperchaotic attractor still remains an unsolved problem. In this paper, we present strong numerical evidence to suggest that the geometric structure of a hyperchaotic attractor can be characterized using a multifractal spectrum with two superimposed components. In other words, apart from developing an extra positive LE, there is also a structural change as a chaotic attractor makes a transition to the hyperchaotic phase and the attractor changes from a simple multifractal to a dual multifractal, equivalent to two inter-mingled multifractals. We argue that a cross-over behavior in the scaling region for computing the correlation dimension is a manifestation of such a structure. In order to support this claim, we present an illustrative example of a synthetically generated set of points in the unit interval (a Cantor set with a variable iteration scheme) displaying dual multifractal spectrum. Our results are also used to develop a general scheme to generate both hyperchaotic as well as high dimensional chaotic attractors by coupling two low dimensional chaotic attractors and tuning a time scale parameter.