Dynamic behavior in a pair of Lorenz systems interacting via positive-negative coupling

A standard assumption when analyzing bidirectionally coupled systems is to consider that the coupling consists of a negative feedback interconnection and also, it is generally assumed that the coupling is symmetric. Here, we investigate, by means of an example, the limit behavior in a pair of bidirectionally coupled systems where the coupling structure is non symmetric and composed by a combination of positive and negative feedback. The example at hand is the well-know Lorenz system. It is demonstrated that, besides synchronization, the coupled systems exhibit emergent behavior characterized by a transition from a synchronized chaotic state to unsynchronized periodic behavior, i.e., each system converges to a different limit cycle. Furthermore, it is shown that the coupling strength and the initial conditions of the systems play a key role in the onset of bistability of solutions. For the analysis, we use the master stability function, the Euclidean distance between trajectories, bifurcation diagrams, and Lyapunov exponents. Additionally, the obtained results are experimentally validated using electronic circuits.