Effects of frequency mismatch on amplitude death in delay-coupled oscillators

Analysis of bifurcation and explosive amplitude death in a pair of oscillators coupled via time-delay connection

Delay-induced amplitude death (AD) has received considerable research interest. Most studies on delay-induced AD investigated the local stability of equilibrium points. The present study examines the global dynamics of delay-induced AD in a pair of identical Stuart-Landau oscillators. Bifurcation diagrams consisting of synchronized periodic orbits and an equilibrium point are used to determine the mechanism of the emergence of delay-induced AD. It is shown that explosive delay-induced AD can occur via a Hopf bifurcation at the equilibrium point and a saddle-node bifurcation of synchronized periodic orbits when the delay time for the connection is continuously varied. The Hopf and saddle-node bifurcation curves in the coupling parameter space clarify the dependence of the coupling parameters on the global dynamics.

Delay-induced amplitude death in multiplex oscillator network with frequency-mismatched layers

The present paper analytically investigates the stability of amplitude death in a multiplex Stuart-Landau oscillator network with a delayed interlayer connection. The network consists of two frequency-mismatched layers, and all oscillators in each layer have identical frequencies. We show that, if the matrices describing the network topologies of each layer commute, then the characteristic equation governing the stability can be reduced to a simple form. This form reveals that the stability of amplitude death in the multiplex network is equally or more conservative than that in a pair of frequency-mismatched oscillators coupled by a delayed connection. In addition, we provide a procedure for designing the delayed interlayer connection such that amplitude death is stable for any commuting matrices and for any intralayer coupling strength. These analytical results are verified through numerical examples. Moreover, we numerically discuss the results for the case in which the commutative property does not hold.