Oscillatory behavior is essential for proper functioning of various physical and biological processes in a wide variety of natural systems. These systems are often composed of an ensemble of interacting oscillatory units; when the interaction occurs through a diffusive manner, the intrinsic oscillations can be suppressed by manifesting two structurally distinct oscillation quenching phenomena: amplitude death (AD) and oscillation death (OD). Generally, AD refers to stabilization of an already existing homogeneous steady state (HSS) to which the coupled units are entrained. In contrast, OD is manifested as a novel inhomogeneous steady state (IHSS), which occurs due to symmetry breaking of the coupled systems, and the distinct units then populate different branches of the same IHSS.

The phenomena of AD and OD can be responsible for a loss of intrinsic dynamics, which may lead to a large degree of degradation in the dynamic performance of physical systems. How to revoke oscillation quenching and efficiently restore rhythmic activity with a general technique is an open and challenging issue. In the paper, we have proposed a rather simple and efficient coupling scheme to revoke both AD and OD to retrieve rhythmic behaviours in diffusively coupled dynamical networks of non-linear oscillators.

By introducing a limiting feedback factor a in the diffusive coupling, we have revealed that this approach is generically robustin revoking AD in coupled Stuart-Landau oscillators under distinct scenarios such as frequency mismatch, delayed coupling with both discrete and distributed time delays, and conjugate and dynamic couplings. We have shown that the OD phenomenon can also be revoked in coupled Stuart-Landau oscillators. Intriguingly, a minimal decrease of a from unity drastically reduces the size of the stable regions of both AD and OD in the parameter space. The effect of a minute deviation of a from unity manifests in switching the stability of the stable HSS and IHSS. Furthermore, we have experimentally confirmed the efficient role of the diffusive control factor a in restoring oscillations by destabilizing AD in a delay-coupled chemical reaction systemwith Ni electrodissolution.

We have corroborated our results by employing the paradigmatic Stuart-Landau oscillator, which represents a normal form describing dynamics near a supercritical Hopf bifurcation. Our findings are expected to hold true for a broad class of coupled non-linear systems near a Hopf bifurcation. In fact, we have confirmed that dynamic activity can be recovered effectively in many other diffusively coupled non-linear oscillators experiencing AD or OD, such as Brusselators, chaotic Lorenz oscillators, Pikovsky-Rabinovich circuit models, synthetic genetic relaxation oscillators and membrane models. The generality of our method has also been successfully validated in diffusively coupled dynamical networks experiencing distinctly different deteriorations of dynamic activity such as partial deaths and aging transition. Thus, the new ingredient a in the coupling serves as a very general framework to strengthen the robustness of dynamic activity in diffusively coupled networks.