Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings

Emerging chimera states under nonidentical counter-rotating oscillators

Frequency plays a crucial role in exhibiting various collective dynamics in the coexisting corotating and counter-rotating systems. To illustrate the impact of counter-rotating frequencies, we consider a network of nonidentical and globally coupled Stuart-Landau oscillators with additional perturbation. Primarily, we investigate the dynamical transitions in the absence of perturbation, demonstrating that the transition from desynchronized state to cluster oscillatory state occurs through an interesting partial synchronization state in the oscillatory regime. Following this, the system dynamics transits to amplitude death and oscillation death states. Importantly, we find that the observed dynamical states do not preserve the parity (P) symmetry in the absence of perturbation. When the perturbation is increased one can note that the system dynamics exhibits a kind of transition which corresponds to a change from incoherent mixed synchronization to coherent mixed synchronization through a chimera state. In particular, incoherent mixed synchronization and coherent mixed synchronization states completely preserve the P symmetry, whereas the chimera state preserves the P symmetry only partially. To demonstrate the occurrence of such partial symmetry-breaking (chimera) state, we use basin stability analysis and discover that partial symmetry breaking exists as a result of the coexistence of symmetry-preserving and symmetry-breaking behavior in the initial state space. Further, a measure of the strength of P symmetry is established to quantify the P symmetry in the observed dynamical states. Subsequently, the dynamical transitions are investigated in the parametric spaces. Finally, by increasing the network size, the robustness of the chimera state is also inspected, and we find that the chimera state is robust even in networks of larger sizes. We also show the generality of the above results in the related reduced phase. model as well as in other coupled models such as the globally coupled van der Pol and Rössler oscillators.

Emergence of multistability and strongly asymmetric collective modes in two quorum sensing coupled identical ring oscillators

The simplest ring oscillator is made from three strongly nonlinear elements repressing each other unidirectionally, resulting in the emergence of a limit cycle. A popular implementation of this scheme uses repressor genes in bacteria, creating the synthetic genetic oscillator known as the Repressilator. We consider the main collective modes produced when two identical Repressilators are mean-field-coupled via the quorum-sensing mechanism. In-phase and anti-phase oscillations of the coupled oscillators emerge from two Andronov-Hopf bifurcations of the homogeneous steady state. Using the rate of the repressor's production and the value of coupling strength as the bifurcation parameters, we performed one-parameter continuations of limit cycles and two-parameter continuations of their bifurcations to show how bifurcations of the in-phase and anti-phase oscillations influence the dynamical behaviors for this system. Pitchfork bifurcation of the unstable in-phase cycle leads to the creation of novel inhomogeneous limit cycles with very different amplitudes, in contrast to the well-known asymmetrical limit cycles arising from oscillation death. The Neimark-Sacker bifurcation of the anti-phase cycle determines the border of an island in two-parameter space containing almost all the interesting regimes including the set of resonant limit cycles, the area with stable inhomogeneous cycle, and very large areas with chaotic regimes resulting from torus destruction and period doubling of resonant cycles and inhomogeneous cycles. We discuss the structure of the chaos skeleton to show the role of inhomogeneous cycles in its formation. Many regions of multistability and transitions between regimes are presented. These results provide new insights into the coupling-dependent mechanisms of multistability and collective regime symmetry breaking in populations of identical multidimensional oscillators.

Additional repulsion reduces the dynamical resilience in the damaged networks

In this paper, we investigate the dynamical robustness of diffusively coupled oscillatory networks under the influence of an additional repulsive link. Such a dynamical resilience property is realized through the aging process of the damaged network of active and inactive oscillators. The aging process is one type of phase transition, mainly appearing at a critical threshold of a fraction of the inactive oscillator node where the mean oscillation amplitude of the entire network suddenly vanishes. These critical fractions of the failure nodes in the network are broadly used as a measure of network resilience. Here, we analytically derived the critical fraction of the aging process in the dynamical network. We find that the addition of the repulsive link enhances the critical threshold of the aging transition of diffusively coupled oscillators, which indicated that the dynamical robustness of the coupled network decreases with the presence of the repulsive interaction. Furthermore, we investigate the dynamical robustness of the network against the number of deteriorating repulsive links. We observed that a certain percentage of the repulsive link is enabled to produce the aging process in the entire network. Finally, the effect of symmetry-breaking coupling and the targeted inactivation process on the dynamical robustness property of damaged networks were investigated. The analytically obtained results are verified numerically in the network of coupled Stuart-Landau oscillators. These findings may help us to better understand the role of the coupling mechanism on the phase transition in the damaged network.

Dynamical effects of breaking rotational symmetry in counter-rotating Stuart-Landau oscillators

Stuart-Landau oscillators can be coupled so as to either preserve or destroy the rotational symmetry that the uncoupled system possesses. We examine some of the simplest cases of such couplings for a system of two nonidentical oscillators. When the coupling breaks the rotational invariance, there is a qualitative difference between oscillators wherein the phase velocity has the same sign (termed co-rotation) or opposite signs (termed counter-rotation). In the regime of oscillation death the relative sense of the phase rotations plays a major role. In particular, when rotational invariance is broken, counter-rotation or phase velocities of opposite signs appear to destabilize existing fixed points, thereby preserving and possibly extending the range of oscillatory behavior. The dynamical "frustration" induced by counter-rotations can thus suppress oscillation quenching when coupling breaks the symmetry.