Chaotic hidden attractor in a fractional order system modelling the interaction between dark matter and dark energy

TL;DR

This paper demonstrates the existence of chaotic hidden attractors in a fractional order system modeling dark matter and dark energy interaction, revealing complex dynamics not connected to equilibria.

Contribution

It is the first to identify chaotic hidden attractors in such a system, combining analytical and numerical methods.

Findings

Chaotic hidden attractors coexist with stable equilibria.

Numerical methods confirm the presence of chaos via Lyapunov exponents.

The attractor is 'hidden', not connected to equilibria in phase space.

Abstract

In this paper the dynamics of a fractional order system modelling the interaction between dark matter and dark energy is analytically and numerically studied. It is shown for the first time that systems modelling the interaction between dark matter and dark energy, chaotic hidden attractors can be present. The chaotic attractor co-exists with two asymptotically stable equilibria. Equilibria of the linearized system exhibit a center-like behavior. The numerical integration is done by means of the Adams-Bashforth-Moulton scheme and the finite Lyapunov exponents are numerically determined with a dedicated Matlab code. The 3D representation of the chaotic hidden attractor reveals the fact it is not connected with the equilibria, being `hidden` somewhere in the considered phase space.