Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles

Author(s)

Medeiros, Everton S. ; Caldas, Ibere L. ; Baptista, Murilo S. ; Feudel, Ulrike

• Document type: Journal article
• Source: SCIENTIFIC REPORTS; v. 7, FEB 9 2017.
• Web of Science Citations: 11

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. (AU)

• FAPESP's process: 11/19296-1 - Nonlinear dynamics
• Grantee: Iberê Luiz Caldas
• Support type: Research Projects - Thematic Grants
• FAPESP's process: 13/26598-0 - Fractal Boundaries between chaos and periodicity in the parameters space.
• Grantee: Everton Santos Medeiros
• Support type: Scholarships in Brazil - Post-Doctorate
• FAPESP's process: 15/20407-3 - Investigating tipping points after-effects
• Grantee: Everton Santos Medeiros
• Support type: Scholarships abroad - Research Internship - Post-doctor