Partial amplitude death in delay-coupled oscillators with complete multipartite graphs

Partial amplitude death is a phenomenon related to both the phase and amplitude in coupled oscillators, where some oscillators converge to their equilibrium points and others oscillate with a particular phase difference. Although partial amplitude death in delay-coupled oscillators has been numerically observed, it has seldom been analyzed theoretically. Here, we analytically investigate partial amplitude death in delay-coupled oscillators with complete multipartite graphs. We analytically derive the amplitude and phase relationship of the oscillators during partial amplitude death. It is shown that various phase-locked patterns of the partial amplitude death state exist in a given network topology. Furthermore, we find that the coupling strength controls the number of oscillators quenched and that the connection delay does not affect the existence of partial amplitude death, but affects its stability. It is shown that the local stability of partial amplitude death can be reduced to that of a time-invariant linear system, which is useful for analyzing the stability of partial amplitude death with a specific phase-locked pattern.