# New method of analysis of the effect of weak colored noise in nonlinear dynamical systems

## Abstract

A systematic method for obtaining the asymptotic behavior of a dynamical system forced by colored noise in the limit of small intensity is developed. It is based on the search of WKB solutions to the Fokker-Planck equation for the joint probability density of the system and noise, in which the perturbation expansion is continued to the first correction beyond the Hamiltonian-Jacobi limit. The method can be applied to noise with correlation time of order unity. It is illustrated on the normal form of a pitchfork bifurcation, where it is pointed out that additive noise can induce a shift of the most probable value. This prediction is confirmed by numerical simulation of the stochastic differential equations.

- Authors:

- Publication Date:

- Research Org.:
- Faculte des Sciences, Universite Libre de Bruxelles, 1050 Bruxelles, Belgium

- OSTI Identifier:
- 6686864

- Resource Type:
- Journal Article

- Journal Name:
- J. Stat. Phys.; (United States)

- Additional Journal Information:
- Journal Volume: 46:1

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; STOCHASTIC PROCESSES; NOISE; FOKKER-PLANCK EQUATION; NONLINEAR PROBLEMS; WKB APPROXIMATION; DIFFERENTIAL EQUATIONS; EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; 657000* - Theoretical & Mathematical Physics

### Citation Formats

```
Altares, V, and Nicolis, G.
```*New method of analysis of the effect of weak colored noise in nonlinear dynamical systems*. United States: N. p., 1987.
Web. doi:10.1007/BF01010340.

```
Altares, V, & Nicolis, G.
```*New method of analysis of the effect of weak colored noise in nonlinear dynamical systems*. United States. doi:10.1007/BF01010340.

```
Altares, V, and Nicolis, G. Thu .
"New method of analysis of the effect of weak colored noise in nonlinear dynamical systems". United States. doi:10.1007/BF01010340.
```

```
@article{osti_6686864,
```

title = {New method of analysis of the effect of weak colored noise in nonlinear dynamical systems},

author = {Altares, V and Nicolis, G},

abstractNote = {A systematic method for obtaining the asymptotic behavior of a dynamical system forced by colored noise in the limit of small intensity is developed. It is based on the search of WKB solutions to the Fokker-Planck equation for the joint probability density of the system and noise, in which the perturbation expansion is continued to the first correction beyond the Hamiltonian-Jacobi limit. The method can be applied to noise with correlation time of order unity. It is illustrated on the normal form of a pitchfork bifurcation, where it is pointed out that additive noise can induce a shift of the most probable value. This prediction is confirmed by numerical simulation of the stochastic differential equations.},

doi = {10.1007/BF01010340},

journal = {J. Stat. Phys.; (United States)},

number = ,

volume = 46:1,

place = {United States},

year = {1987},

month = {1}

}

DOI: 10.1007/BF01010340

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