Adiabatic reduction near a bifurcation in stochastically modulated systems

Details Adiabatic reduction near a bifurcation in stochastically modulated systems Adiabatic reduction near a bifurcation in stochastically modulated systems

1997-12-03

arxiv.org/abs/patt-sol/9712001v1

Phys. Rev. E 57, 5036 (1998).

Nonlinear Sciences

Pattern Formation and Solitons

RevTeX, 19 pages and 16 figures

Scientific paper

10.1103/PhysRevE.57.5036

We re-examine 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. Stochastic resonance 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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