Inhomogeneous to homogeneous dynamical states through symmetry breaking dynamics

We explore the dynamical transitions facilitated by the symmetry breaking dynamical states in a directly coupled limit-cycle oscillators, when the oscillators are interacting through attractive and repulsive couplings. Initially, we show that the transition from out-of-phase synchronized (OPS) state to in-phase synchronized (IPS) state is mediated by the onset of symmetry breaking oscillatory (SBO) state using two coupled systems. Similarly, we find the emergence of symmetry breaking oscillation death (AOD) state when the system transits from oscillation death (OD) state to nontrivial amplitude death (NAD) state. Further, we deduce the analytical stability conditions for all the observed dynamical states, namely the IPS, OPS, NAD, OD, including SBO and AOD states. Additionally, the investigation on symmetry breaking dynamics in a ring of unidirectionally coupled oscillators reveals that the breaking of symmetry leads to the traveling chimera (TC) and symmetry breaking incoherent oscillation death states. We also report that the transient nature of TC state obeys a power-law relation near the stable TC region as a function of the coupling strength. From the results, it is evident that the trade-off between the attractive and the repulsive couplings induces the symmetry breaking dynamics in the entire range of the system frequency.