Download:
|
by Nils Berglund, Barbara Gentz
http://dpwww.epfl.ch/instituts/ipt/berglund/geom.ps.gz
Add To MetaCart
Abstract:
We consider slow{fast systems of dierential equations, in which both the slow and fast variables are perturbed by additive noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the reduced system, the results cover time spans which can be exponentially long in the noise intensity squared (that is, up to Kramers ' time). We give exponentially small upper and lower bounds on the probability of exceptional paths. If the slow manifold contains bifurcation points, we show similar concentration properties for the fast variables corresponding to non-bifurcating modes. We also give conditions under which the system can be approximated by a lower-dimensional one, in which the fast variables contain only bifurcating modes.
Citations
|
175
|
Random perturbations of dynamical systems
– Freidlin, Wentzell
- 1984
|
|
127
|
Random dynamical systems
– Arnold
- 1998
|
|
107
|
Introduction to Matrix Analysis
– Bellman
- 1970
|
|
81
|
Geometric singular perturbation theory for ordinary differential equations
– Fenichel
- 1979
|
|
40
|
1995] Geometric Singular Perturbation Theory
– Jones
- 1994
|
|
22
|
Attractors for random dynamical systems, Probab. Theory Related Fields 100
– Crauel, Flandoli
- 1994
|
|
7
|
Asymptotic series and exit time probabilities
– Fleming, James
- 1992
|
|
7
|
Slowly varying jump and transition phenomena associated with algebraic bifurcation problems
– Haberman
- 1979
|
|
7
|
Exchange of stabilities in autonomous systems
– Lebovitz, Schaar
- 1975
|
|
7
|
Dierential Equations with Small Parameters and Relaxation Oscillations
– Rozov
- 1980
|
|
7
|
Systems of dierential equations containing small parameters
– Tihonov
- 1952
|
|
6
|
Pathwise description of dynamic pitchfork bifurcations with additive noise
– Berglund, Gentz
- 2002
|
|
6
|
The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point
– Kifer
- 1981
|
|
5
|
A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential
– Berglund, Gentz
- 2000
|
|
5
|
On the exponential exit law in the small parameter exit problem
– Day
- 1983
|
|
4
|
Petites perturbations aleatoires des systemes dynamiques: developpements asymptotiques
– Azencott
- 1985
|
|
4
|
Invariant attracting sets of nonlinear stochastic dierential equations
– Schmalfu
- 1989
|
|
3
|
Metastability in reversible diusion processes I. Sharp asymptotics for capacities and exit times, Preprint WIAS-767
– Bovier, Eckho, et al.
- 2002
|
|
3
|
On stable oscillations and equilibriums induced by small noise
– Freidlin
- 2001
|
|
3
|
Asymptotic behavior of solutions of systems of dierential equations with a small parameter in the derivatives of highest order
– Pontryagin
- 1957
|
|
3
|
Approximate solution of a system of ordinary dierential equations involving a small parameter in the derivatives
– Pontryagin, Rodygin
- 1960
|
|
2
|
Metastability in reversible diusion processes II. Precise asymptotics for small eigenvalues
– Bovier, Gayrard, et al.
- 2002
|
|
2
|
On the exit law from saddle points. Stochastic Process
– Day
- 1995
|
|
2
|
Application of A. M. Lyapunov's theory of stability to the theory of dierential equations with small coecients in the derivatives
– Gradsten
- 1953
|
|
1
|
The eect of additive noise on dynamical hysteresis. Nonlinearity
– Berglund, Gentz
- 2002
|
|
1
|
Persistence of stability loss for dynamical bifurcations I. Dierential Equations
– Neshtadt
- 1987
|
|
1
|
Persistence of stability loss for dynamical bifurcations II. Dierential Equations
– Neshtadt
- 1988
|
