Complex amplitude chimeras and amplitude-mediated chimeras in networks of birhythmic van der Pol oscillators

The chimera state is an amazing partial synchronization state characterized by coexisting coherent and incoherent behavior. Chimera states have been intensively studied in networks of monostable systems. However, their investigation in coupled multistable systems is a new subject. In this paper, we investigate the chimera phenomenon in a network of a two-variable van der Pol model with extended nonlinearity characterized by a special form of bistability known as birhythmicity. Under specially prepared initial conditions, the network of locally coupled birhythmic van der Pol oscillators exhibits diverse amplitude chimera and amplitude-mediated chimera states. Interestingly, we observe novel transient and stable amplitude chimera states whose spatial profiles of the shift of oscillations center of mass from origin are non-smooth. In the incoherent parts, these spatial profiles switch between two curves which correspond to two pairs of symmetric steady states. With the help of a bifurcation analysis of a reduced model of the network, we highlight the origin of these two pairs of steady states which are also at the basis of a four-level oscillation death phenomenon. Besides, we explore the possibility of occurrence of these complex patterns in more complex networks, including a network with nonlocal coupling, a small-world network, and a completely random network. We find that the chimera states tend to disappear when the coupling range of the nonlocal coupling increases. On the other hand, other forms of stable amplitude chimera patterns appear in the considered small-world network, among which a peculiar form which involves the two limit cycles of the birhythmic model. Furthermore, a strong randomness in the coupling configuration leads to the suppression of all the chimera states observed in the regular and small-world networks.