Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles

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

Research output: Contribution to journalArticle

12 Citations (Scopus)
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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.
Original languageEnglish
Article number42351
JournalScientific Reports
Volume7
Publication statusPublished - 9 Feb 2017

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trapping
cycles
critical point
ecology
twisting
transition points
ghosts
dynamical systems
climate
trajectories
oscillations
thresholds
shift

Keywords

  • natural hazards
  • phase transitions and critical phenomena

Cite this

Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles. / Medeiros, Everton S.; Caldas, Ibere L.; Baptista, Murilo S.; Feudel, Ulrike.

In: Scientific Reports, Vol. 7, 42351, 09.02.2017.

Research output: Contribution to journalArticle

Medeiros, Everton S. ; Caldas, Ibere L. ; Baptista, Murilo S. ; Feudel, Ulrike. / Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles. In: Scientific Reports. 2017 ; Vol. 7.
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AU - Caldas, Ibere L.

AU - Baptista, Murilo S.

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N1 - We would like to thank the partial support of this work by the Brazilian agencies FAPESP (processes: 2011/19296-1, 2013/26598-0, and 2015/20407-3), CNPq and CAPES. MSB acknowledges EPSRC Ref. EP/I032606/1.

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AB - 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.

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