Geometric singular perturbation theory for stochastic differential equations

Abstract

We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by 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 obtain 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.

43 pages. Published version. Remarks added, minor corrections

References in corpus (3)

• Pathwise description of dynamic pitchfork bifurcations with additive noise
• The effect of additive noise on dynamical hysteresis
• A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential

Cited by in corpus (18)

• A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics
• A mathematical framework for critical transitions: normal forms, variance and applications
• Normal form transforms separate slow and fast modes in stochastic dynamical systems
• Long-Term Stability Analysis of Power Systems with Wind Power Based on Stochastic Differential Equations: Model Development and Foundations
• Hunting French Ducks in a Noisy Environment
• Analytical Study of the Impacts of Stochastic Load Fluctuation on the Dynamic Voltage Stability Margin Using Bifurcation Theory
• Stochastic dynamics on slow manifolds
• A Framework for Dynamic Stability Analysis of Power Systems with Volatile Wind Power
• From random Poincaré maps to stochastic mixed-mode-oscillation patterns
• Escape Rates in a Stochastic Environment with Multiple Scales
• Noise-induced tipping under periodic forcing: preferred tipping phase in a non-adiabatic forcing regime
• Heterogeneous Population Dynamics and Scaling Laws near Epidemic Outbreaks
• Stochastic resonance in stochastic PDEs
• The effect of classical noise on a quantum two-level system
• Spectral theory for random Poincaré maps
• Discovery of slow variables in a class of multiscale stochastic systems via neural networks
• Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus
• A General View on Double Limits in Differential Equations