Revival of Oscillations from Explosive Death Under Mean-Field Attractive–Repulsive Coupling

The restoration and maintenance of oscillatory rhythms are integrated to the regular operation of biological and physical processes. In this paper, we explore the revival of oscillations, specifically focusing on the phenomenon of explosive Nontrivial Amplitude Death (NAD) that occurs within N van der Pol oscillators coupled through mean-field attractive–repulsive interactions. By incorporating discrete and distributed delays into the repulsive mean-field, respectively, we provide theoretical and numerical evidence that the explosive NAD can be effectively destabilized, leading to the restoration of the oscillatory behavior. By utilizing Lyapunov stability, we theoretically determine the boundaries of the oscillation region recovered from NAD. For discrete delay, adjusting the magnitude of delay can result in a substantial reduction of the NAD region within the parameter space. Under a specific delay, the NAD region completely becomes an oscillation state. Moreover, in the scenario of uniformly distributed delay, the presence of delay also contributes to a decrease of the NAD area, and as the distribution width increases, the revived oscillation region gradually contracts compared to the case with discrete delay. Importantly, the results from the numerical simulations are in agreement with the theoretical analysis. Our framework establishes a theoretical foundation for comprehending the regeneration of oscillation in mean-field attractive–repulsive coupled systems.