[Paper Review] Slow Manifolds for Multi-Time-Scale Stochastic Evolutionary Systems

Hongbo Fu, Xianming Liu+1 more|arXiv (Cornell University)|May 2, 2012

Stability and Controllability of Differential Equations22 references3 citations

TL;DR

This paper establishes the existence of an exponentially tracking random invariant manifold for multi-time-scale stochastic evolutionary systems in infinite dimensions, using the Lyapunov-Perron method. It proves that as the fast-slow time-scale ratio ε → 0, this manifold converges to a slow manifold that captures the long-term dynamics of the reduced system, with applications to coupled parabolic-hyperbolic and hyperbolic-hyperbolic SPDEs.

ABSTRACT

This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations. Under suitable conditions, it is proved that an exponentially tracking random invariant manifold exists, eliminating the fast motion for this coupled system. It is further shown that if the scaling parameter tends to zero, the invariant manifold tends to a slow manifold which captures long time dynamics. As examples the results are applied to a few systems of coupled parabolic-hyperbolic partial differential equations, coupled parabolic partial differential-ordinary differential equations, and coupled hyperbolic-hyperbolic partial differential equations.

Motivation & Objective

• To analyze the long-time dynamics of infinite-dimensional stochastic systems with widely separated time scales.
• To establish the existence of a random invariant manifold that eliminates fast motion in coupled fast-slow SPDEs.
• To demonstrate that this manifold asymptotically approaches a slow manifold as the scaling parameter ε tends to zero.
• To extend the theory of invariant manifolds to stochastic systems with additive noise and non-Markovian noise structures.
• To provide a rigorous reduction framework for complex multiscale SPDEs via invariant manifold theory.

Proposed method

• Applies the Lyapunov-Perron method to construct a random invariant manifold for a coupled system of fast-slow SPDEs.
• Imposes conditions ensuring the Lipschitz constant of the fast component's nonlinearity is small relative to the decay rate of the linear operator A.
• Uses a cut-off technique to extend results to locally Lipschitz nonlinearities by restricting the dynamics to bounded sets.
• Defines the invariant manifold as the graph of a random function h^ε(ω, Y₀) mapping slow variables to fast variables.
• Employs the random dynamical system framework generated by the stochastic flow, with noise modeled as additive Brownian motion.
• Proves exponential tracking by showing trajectories converge uniformly to the manifold at an exponential rate in time.

Experimental results

Research questions

• RQ1 Does a random invariant manifold exist for a coupled system of fast-slow stochastic evolutionary equations in infinite dimensions?
• RQ2 Can this manifold be shown to possess an exponential tracking property for small ε > 0?
• RQ3 What happens to the invariant manifold as the time-scale separation parameter ε approaches zero?
• RQ4 Can the limiting object be identified as a slow manifold that captures the long-term dynamics?
• RQ5 How does the theory extend to systems with locally Lipschitz nonlinearities?

Key findings

• For sufficiently small ε > 0, there exists a random invariant manifold M^ε with an exponential tracking property, reducing the full system to a lower-dimensional effective equation.
• The manifold M^ε converges to a slow manifold M⁰ as ε → 0, which governs the long-time behavior of the system.
• The existence of the invariant manifold is guaranteed under the condition that the Lipschitz constant of the fast nonlinearity is less than the decay rate of the linear operator A.
• The method applies to coupled parabolic-hyperbolic, parabolic-ODE, and hyperbolic-hyperbolic SPDE systems, as demonstrated in explicit examples.
• For locally Lipschitz nonlinearities, a local random invariant manifold exists via a cut-off procedure, valid within bounded regions.
• The convergence of M^ε to M⁰ is established in the sense of the random dynamical system's asymptotic behavior, with uniform convergence in time.