Mixed synchronization in multiplex networks of counter-rotating oscillators

In this study, we introduce a novel concept known as "mixed synchronization" within the context of two indirectly coupled counter-rotating oscillators. Our investigation begins by exploring the dynamics of two co-rotating oscillators when they are connected through a counter-rotating relay oscillator. Remarkably, this coupling mechanism leads to the emergence of what we term "relay synchronization," which signifies complete synchronization between the outer oscillators. This intriguing phenomenon lays the foundation for our subsequent exploration. We delve deeper into the dynamics of counter-rotating oscillators and discover that when two such oscillators are linked through relay oscillators, a distinct form of synchronization arises, which we aptly term "relay mixed synchronization." Notably, this reveals that even if counter-rotating oscillators are part of a larger network and indirectly coupled through a simple diffusive mechanism, it is possible to induce mixed synchronization. We extend the same innovative concept to the realm of multiplex networks, which consist of a finite number of counter-rotating oscillators in each layers and we explore both intralayer and interlayer mixed synchronization states. We verify the mixed synchronization state by master stability function analysis. We numerically verify the results by considering limit cycle and chaotic oscillators. This extension of our concept to multiplex networks showcases the versatility and applicability of mixed synchronization in diverse scenarios.