State-dependent vulnerability of synchronization.

A state-dependent vulnerability of synchronization is shown to exist in a complex network composed of numerically simulated electronic circuits. We demonstrate that disturbances to the local dynamics of network units can produce different outcomes to synchronization depending on the current state of its trajectory. We address such state dependence by systematically perturbing the synchronized system at states equally distributed along its trajectory. We find the states at which the perturbation desynchronizes the network to be complicatedly mixed with the ones that restore synchronization. Additionally, we characterize perturbation sets obtained for consecutive states by defining a safety index between them. Finally, we demonstrate that the observed vulnerability is due to the existence of an unstable chaotic set in the system's state space.

Publisher Correction: Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles.

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Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles.

Nonlinear dynamical systems may be exposed to tipping points, critical thresholds at which small changes in the external inputs or in the system's parameters abruptly shift the system to an alternative state with a contrasting dynamical behavior. While tipping in a fold bifurcation of an equilibrium is well understood, much less is known about tipping of oscillations (limit cycles) though this dynamics are the typical response of many natural systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate sciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show that, when limit cycles are considered, a transient structure, so-called channel, plays a fundamental role in the transition. Specifically, we demonstrate that trajectories crossing such channel conserve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifurcation of cycles. As a consequence, this channel acts like a "ghost" of the limit cycle destroyed in the critical transition and instead of the expected abrupt transition we find a smooth one. This smoothness is also the reason that it is difficult to precisely determine the transition point employing the usual indicators of tipping points, like critical slowing down and flickering.