Simplest symmetric chaotic flows: the strange case of asymmetry in Master Stability Function

In this research, we investigate the existence of symmetry in the simplest three-dimensional chaotic flows with unique features. We search the simplest Sprott chaotic systems, systems with no equilibrium, stable equilibrium, and systems with the line, curve, and surface equilibrium. We show that some of such systems are symmetric systems. Also, only a few have coexisting symmetric attractors. Moreover, we study the synchronization of these symmetric systems to understand the collective behavior of the network of such systems. We compute the Master Stability Function, which provides a necessary condition for synchronization. We consider the linear coupling function in different one-component schemes. It is observed that the synchronization in these systems, has no relation with the coupling of symmetric variables. Furthermore, the results show that the attractors may have different Master Stability Functions for the systems with coexisting symmetric attractors.