Publication

Abstract : In this article, we investigate the dynamical robustness in a network of Van der Pol oscillators. In particular, we consider a network of diffusively coupled Van der Pol oscillators to explore the aging transition phenomena. Our investigation reveals that the route to aging transition in a network of Van der Pol oscillator is different from that of typical sinusoidal oscillators such as Stuart–Landau oscillators. Unlike sinusoidal oscillators, the order parameter does not follow smooth second-order phase transition. Rather, we observe an abnormal phase transition of the order parameter due to the sudden appearance of unbounded trajectories at a critical point. We provide detailed bifurcation analysis of such an abnormal phase transition. We show that the boundary crisis of a limit-cycle oscillator is at the helm of such an unusual discontinuous path of aging transition. The network of coupled oscillators is an efficient model to explore various self-organizing activities of complex systems in the disciplines of physics, biology, and engineering. Having robust oscillatory dynamics is a prerequisite for the normal functioning of such complex systems. The rhythmic activities of such a large-scale system should be resilient against any local degradation or deterioration. In the absence of any rhythmic activities, the regular functioning of many natural and man-made systems may face severe disruption and this emergent phenomenon is known as aging transition. Recently, the study of the robustness of the oscillatory dynamics of a complex dynamical system has become an emerging area of research. The basic framework of such an investigation involves a network of oscillatory nodes where some components of nodes are functionally degraded but not removed. Until now, the investigation of dynamical robustness was mainly limited to coupled Stuart–Landau oscillators, which have typical sinusoidal oscillation. However, there are many natural systems that can be modeled by a network non-sinusoidal oscillators such as Van der Pol oscillator. Van der Pol oscillator has been used to model many real-life systems that includes neuronal activities in the brain cortex and cardiac rhythms. In this article, we have studied the aging transition in a network of Van der Pol oscillators. Our investigation reveals some interesting phenomena. The order parameter that represents the dynamical activities in the network goes through an abnormal phase transition and suddenly blows up to infinity. We provide detailed bifurcation analysis for such an abnormal phase transition. A blue-sky catastrophe of a limit-cycle oscillator is responsible for the unbounded dynamics of the oscillators. Our results provide significant insights into the aging transition of complex dynamical systems where individual nodes represent a Van der Pol oscillator.