Trapping Phenomenon Attenuates Tipping Points for Limit Cycles. Medeiros, E. S., Caldas, I. L., Baptista, M. S., & Feudel, U. arXiv preprint arXiv:1610.01656, 2016. 00000
Trapping Phenomenon Attenuates Tipping Points for Limit Cycles [link]Paper  abstract   bibtex   
Nonlinear dynamical systems may be exposed to tipping point s, critical thresholds at which small changes in the external inputs or in the systems parame ters 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 t ipping of oscillations (limit cycles) though this dynamics are the typical response of many natura l systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate s ciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show t hat, when limit cycles are considered, a transient structure, so-called channel, plays a fundamen tal role in the transition. Specifically, we demonstrate that trajectories crossing such channel conse rve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifu rcation of cycles. As a consequence, this channel acts like a “ghost of the limit cycle destroyed i n the critical transition and instead of the expected abrupt transition we find a smooth one. This sm oothness is also the reason that it is difficult to precisely determine the transition point em ploying the usual indicators of tipping points, like critical slowing down and flickering.
@article{medeiros_trapping_2016,
	title = {Trapping {Phenomenon} {Attenuates} {Tipping} {Points} for {Limit} {Cycles}},
	url = {https://arxiv.org/abs/1610.01656},
	abstract = {Nonlinear dynamical systems may be exposed to tipping point
s, critical thresholds at which
small changes in the external inputs or in the systems parame
ters 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 t
ipping of oscillations (limit cycles)
though this dynamics are the typical response of many natura
l systems to a periodic external
forcing, like e.g. seasonal forcing in ecology and climate s
ciences. We provide a detailed analysis of
tipping phenomena in periodically forced systems and show t
hat, when limit cycles are considered,
a transient structure, so-called channel, plays a fundamen
tal role in the transition. Specifically, we
demonstrate that trajectories crossing such channel conse
rve, for a characteristic time, the twisting
behavior of the stable limit cycle destroyed in the fold bifu
rcation of cycles. As a consequence,
this channel acts like a “ghost of the limit cycle destroyed i
n the critical transition and instead
of the expected abrupt transition we find a smooth one. This sm
oothness is also the reason that
it is difficult to precisely determine the transition point em
ploying the usual indicators of tipping
points, like critical slowing down and flickering.},
	urldate = {2016-12-29},
	journal = {arXiv preprint arXiv:1610.01656},
	author = {Medeiros, Everton S. and Caldas, Iberê L. and Baptista, Murilo S. and Feudel, Ulrike},
	year = {2016},
	note = {00000},
	keywords = {collapse, early-warning-signals},
	file = {Medeiros et al. - 2016 - Trapping Phenomenon Attenuates Tipping Points for .pdf:C\:\\Users\\rsrs\\Documents\\Zotero Database\\storage\\DGECE9CR\\Medeiros et al. - 2016 - Trapping Phenomenon Attenuates Tipping Points for .pdf:application/pdf}
}
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