Chimera states in fractional-order coupled Rayleigh oscillators

Chimera states, characterized by the coexistence of synchronized and desynchronized oscillators, have been extensively described via integer differential equations over the past few decades. However, it is still unclear whether and under which conditions chimera states will appear in terms of the coupled systems involving fractional derivatives. In this work, we investigate the chimera states of Rayleigh oscillators in the fractional-order systems with attractive and repulsive couplings. Firstly, we introduce the repulsive control factor and show the various transition routes from the desynchronized state to the incoherent oscillation death by altering this factor and the fractional-order derivatives. Then, it is demonstrated that as the coupling strength varies, a vast repertoire of interesting collective behaviors will emerge, such as synchronization, solitary state, traveling wave, traveling chimera, oscillatory cluster, imperfect traveling chimera state, imperfect amplitude chimera state, and chimera death. Finally, we discuss the effect of attractive and repulsive couplings and find that decreasing the repulsive control factor can reduce the repulsive interaction, allowing the attractive coupling to dominate and promote the emergence of the coherent state.