Hyperchaos & labyrinth chaos: Revisiting Thomas–Rössler systems

Abstract

We consider a multi-dimensional extension of the Thomas-Rössler (TR) systems, that was inspired by Thomas' earlier work on biological feedback circuits, and we report on our first results that shows its ability to sustain a spatio-temporal behaviour, reminiscent of chimera states. The novelty here being that its underlying mechanism is based on "chaotic walks" discovered by René Thomas during the course of his investigations on what he called Labyrinth Chaos. We briefly review the main properties of TR systems and their chaotic and hyperchaotic dynamics and discuss the simplest way of coupling, necessary for this spatio-temporal behaviour that allows the emergence of complex dynamical behaviours. We also recall René Thomas' memorable influence and interaction with the authors as we dedicate this work to his memory.

A New Fractional-Order Hyperchaotic System and Its Adaptive Tracking Control

“…A chaotic system with two or more positive Lyapunov exponents is called the hyperchaotic system. Compared with chaos, the attractor orbits of the hyperchaotic system are separated in more directions [5][6][7][8][9][10]. As thus, the randomness and unpredictability about the hyperchaotic system are also stronger, which makes the hyperchaotic system have a broader development prospect in image encryption and secure communication.…”

Atangana–seda Numerical Scheme for Labyrinth Attractor With New Differential and Integral Operators

“…Basios and Antonopoulos analyzed the main properties of Labyrinth systems and their chaotic and hyperchaotic dynamics. 14 Berre et al proposed a numerical experiment which displays a well defined bifurcation from parallel rolls to labyrinths as the more stable state for the nonvariational model. 15 Xin et al proposed a fractional model of Labyrinth chaos and they used modified Adams-Bashforth-Moultan numerical scheme for the solution of this model.…”

Labyrinth chaos: Revisiting the elegant, chaotic, and hyperchaotic walks

“…In their venture 46,47,49 , Rössler and Thomas proposed a class of continuous-flow dynamical systems, called Thomas-Rössler (TR) systems 6 . Their motive was to in-vestigate what kind of nonlinearity is necessary to support chaos and hyperchaos in the simplest possible way.…”

“…What Rössler and Thomas called labyrinth chaos and "Arabesques" 4,6,47 drew wide attention as examples of elegant chaos, also discussed in 41 . Here, the term "elegant" is understood in the sense of Poincaré's demand for beauty and simplicity in expressing mathematical ideas, something that Rössler and Thomas' work surely adheres to.…”

“…Rössler and co-workers have also demonstrated that labyrinth chaos and labyrinth walks are hyperchaotic systems in more than three dimensions, showing the truth of condition (iv) as the necessary condition for the existence of hyperchaos 47 . Amazingly, these systems still continue to surprise us with their elegance, beauty and combination of properties: As it has been reported recently in 6 , arrays of coupled labyrinth chaos systems exhibit stereotypical, chimera-like states, reminiscent of chimera states observed in coupled Kuramoto oscillators 2,6,24,32 and in other systems 18,21,33,36,37 . These results emphasise further its significance as an example of an elegant dynamical system with a repertoire of rich properties, among which, hyperchaoticity is prominent.…”