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Symmetry-breaking oscillation death in nonlinear oscillators with time-delayed coupling
The collective behavior in coupled nonlinear systems and networks of oscillators is of great current interest. Besides various synchronization patterns, special attention has recently been paid to oscillation suppression. There are two types of oscillation quenching known in the literature: amplitude and oscillation death. Although the distinction between amplitude death and oscillation death is not always obvious, the underlying mechanisms are crucially different. Amplitude death appears as a result of the stabilization of an already existing steady state that is unstable in the absence of coupling. On the contrary, oscillation death occurs due to a newly created stationary state which is a result of symmetry breaking of the homogeneous steady state. Therefore, amplitude death is represented by a symmetric homogeneous steady state, whereas oscillation death is characterized by an inhomogeneous steady state. It has also been shown that amplitude death can occur in time-delayed systems. In contrast to this, the relation between symmetry breaking of the system in presence of time-delay, and thus the formation of inhomogeneous steady state, i.e. oscillation death, under these conditions has not been tackled. In the present work we investigate oscillation death for time-delayed coupling. Using a paradigmatic model of coupled Stuart-Landau oscillators, we define the coupling structures for which symmetry breaking of the homogeneous steady state occurs, and study how oscillation death can be controlled by introducing time-delay in the system. We also discuss the importance of time-delayed amplitude death and oscillation death from application viewpoint in physics and biology. Oscillation death is especially relevant for biological systems, since it provides a mechanism of cellular differentiation.