Effects of coupling on extremely multistable fractional-order systems

Abstract

Multistability is a phenomenon in which a dynamical system possesses multiple attractors within the same set of parameters. This study explores the effects of coupling on the dynamics of a recently introduced 2D single-neuron neural network model. The system is in its original form bistable, but its extension to include fractional-order derivatives yields an unprecedented level of complexity that is characterized by extreme multistability. We provide a comprehensive analysis of the behavior of this system when coupled. We find a diverse range of attractors for two coupled systems emerging from different initial conditions, and we elucidate the transition from periodic to chaotic behavior as the coupling strength increases. We also show that the synchronization dynamics in this case ranges from anti-phase synchronization under weak coupling to complete synchronization under stronger coupling. Finally, we study the collective behavior of several coupled extremely multistable fractional-order systems, which compared to bistable systems exhibit more complexity and uncertainty as they oscillate between different attractors, or even between emerging attractors.