Adiabatic reduction near a bifurcation in stochastically modulated systems

François Drolet, Jorge Viñals

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

We reexamine the procedure of adiabatic elimination of fast relaxing variables near a bifurcation point when some of the parameters of the system are stochastically modulated. Approximate stationary solutions of the Fokker-Planck equation are obtained near threshold for the pitchfork and transcritical bifurcations. Correlations between fast variables and random modulation may shift the effective bifurcation point by an amount proportional to the intensity of the fluctuations. We also find that fluctuations of the fast variables above threshold are not always Gaussian and centered around the (deterministic) center manifold as was previously believed. Numerical solutions obtained for a few illustrative examples support these conclusions.

Original languageEnglish (US)
Pages (from-to)5036-5043
Number of pages8
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume57
Issue number5
DOIs
StatePublished - Jan 1 1998

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Bifurcation
Bifurcation Point
Fluctuations
Transcritical Bifurcation
Pitchfork Bifurcation
thresholds
Center Manifold
Fokker-Planck equation
Fokker-Planck Equation
Stationary Solutions
Elimination
elimination
Modulation
Directly proportional
Numerical Solution
modulation
shift

Cite this

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AU - Viñals, Jorge

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AB - We reexamine the procedure of adiabatic elimination of fast relaxing variables near a bifurcation point when some of the parameters of the system are stochastically modulated. Approximate stationary solutions of the Fokker-Planck equation are obtained near threshold for the pitchfork and transcritical bifurcations. Correlations between fast variables and random modulation may shift the effective bifurcation point by an amount proportional to the intensity of the fluctuations. We also find that fluctuations of the fast variables above threshold are not always Gaussian and centered around the (deterministic) center manifold as was previously believed. Numerical solutions obtained for a few illustrative examples support these conclusions.

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