Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles

Abstract

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.

Cascading regime shifts within and across scales

“…Time correlation is debated given that spatial heterogeneity can break the synchrony induced by the sharing of drivers (17 ), meaning that contextual settings matter for such correlations to emerge. Spatial heterogeneity is also attributed as a mechanism that can smooth out critical transitions and soften their abruptness (24,25 ). Yet, with or without masking mechanisms, identifying common drivers is useful for designing management strategies that target bundles of drivers instead of well-studied variables independently, increasing the chances that managers will avoid several regime shifts under the influence of the same sets of drivers (13,26 ).…”

Antimonotonicity, Crisis and Multiple Attractors in a Simple Memristive Circuit

“…An electronic circuit to reproduce the proposed memristive circuit was designed, and experiments were conducted to verify the results obtained from the numerical simulations. Circuits with coexistence of attractors and interior crisis can be used to understand tipping points [51]. Those circuits that present antimonotonicity and are simultaneously simple in electronic architecture are specially interesting to construct electronic based circuits for chaos-based cryptography with expanded key spaces, [52] or to create fast and light (low computation power required) chaos-based cryptosystems, such as in Ref.…”

Entropy-based generating Markov partitions for complex systems

“…For systems with noise, Ref. 32 has shown that important dynamical characteristics do not fully reveal the status of the system. Another well-studied case where the tipping point happens, is the existence of multi-stability, i.e., the destruction of one attractor or the complete destruction of the oscillatory behaviour (oscillation death).…”