Chaotic attractors that exist only in fractional-order case

Abstract

Introduction: Studying chaotic dynamics in fractional- and integer-order dynamical systems has let researchers understand and predict the mechanisms of related non-linear phenomena.

Objectives: Phase transitions between the fractional- and integer-order cases is one of the main problems that have been extensively examined by scientists, economists, and engineers. This paper reports the existence of chaotic attractors that exist only in the fractional-order case when using the specific selection of parameter values in a new hyperchaotic (Matouk's) system.

Methods: This paper discusses stability analysis of the steady-state solutions, existence of hidden chaotic attractors and self-excited chaotic attractors. The results are supported by computing basin sets of attractions, bifurcation diagrams and the Lyapunov exponent spectrum. These tools verify the existence of chaotic dynamics in the fractional-order case; however, the corresponding integer-order counterpart exhibits quasi-periodic dynamics when using the same choice of initial conditions and parameter set. Projective synchronization via non-linear controllers is also achieved between drive and response states of the hidden chaotic attractors of the fractional Matouk's system.

Results: Dynamical analysis and computer simulation results verify that the chaotic attractors exist only in the fractional-order case when using the specific selection of parameter values in the Matouk's hyperchaotic system.

Conclusions: An example of the existence of hidden and self-excited chaotic attractors that appears only in the fractional-order case is discussed. So, the obtained results give the first example that shows chaotic states are not necessarily transmitted between fractional- and integer-order dynamical systems when using a specific selection of parameter values. Chaos synchronization using the hidden attractors' manifolds provides new challenges in chaos-based applications to technology and industrial fields.

Keywords: Chaos; Fractional-order; Integer-order; Projective synchronization; Quasi-periodic attractor.