Bifurcations in fluctuating systems: The center-manifold approach

• DOI: 10.1007/bf01018837
• Publication Date: 2005-01-19T16:45:57Z
• Authors: E. Knobloch; K. A. Wiesenfeld
• Keywords: 0103 physical sciences; 01 natural sciences

Bifurcations in fluctuating dynamical systems are studied using the ideas of center-manifold reduction. The method provides not only a systematic procedure for the reduction of the system to a small number of variables-but also a classification scheme for the different kinds of dynamical behavior possible near bifurcation points. The joint probability density factorizes into a stationary Gaussian densityp(v/u) in the fast variablesv, and a time-dependent densityP(u, f) in the slow variablesu describing the dynamics on the center manifoldv=v0 (u). P(u, t) obeys a reduced Fokker-Planck equation that can be written in a normal form by means of local nonlinear transformations. Both additive and multiplicative white noise are considered, as is colored noise. The results extend and formalize Haken's concept of adiabatic elimination of fast variables.