Early-Warning Signs for Pattern-Formation in Stochastic Partial Differential Equations
arXiv:1403.5920 · doi:10.1016/j.cnsns.2014.09.019
Abstract
There have been significant recent advances in our understanding of the potential use and limitations of early-warning signs for predicting drastic changes, so called critical transitions or tipping points, in dynamical systems. A focus of mathematical modeling and analysis has been on stochastic ordinary differential equations, where generic statistical early-warning signs can be identified near bifurcation-induced tipping points. In this paper, we outline some basic steps to extend this theory to stochastic partial differential equations with a focus on analytically characterizing basic scaling laws for linear SPDEs and comparing the results to numerical simulations of fully nonlinear problems. In particular, we study stochastic versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a scaling law of the covariance operator in a regime where linearization is expected to be a good approximation for the local fluctuations around deterministic steady states. We compare these results to direct numerical simulation, and study the influence of noise level, noise color, distance to bifurcation and domain size on early-warning signs.
Published in Communications in Nonlinear Science and Numerical Simulation (2014)
References in corpus (20)
- Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system
- A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics
- Early warning signals for critical transitions: A generalized modeling approach
- Spatiotemporal Stochastic Resonance in the Swift-Hohenberg Equation
- A mathematical framework for critical transitions: normal forms, variance and applications
- Rough evolution equations
- Spatial Coherence Resonance near Pattern-Forming Instabilities
- Climate tipping as a noisy bifurcation: a predictive technique
- Pathwise description of dynamic pitchfork bifurcations with additive noise
- Hunting French Ducks in a Noisy Environment
- Noise, coherent fluctuations, and the onset of order in an oscillated granular fluid
- Early warning signs for saddle-escape transitions in complex networks
- Modulation Equations: Stochastic Bifurcation in Large Domains
- Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes
- Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids
- Critical Transitions in Social Network Activity
- Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts
- Numerical Continuation and SPDE Stability for the 2D Cubic-Quintic Allen-Cahn Equation
- Stochastic PDEs: domain formation in dynamic transitions
- On spatial and temporal multilevel dynamics and scaling effects in epileptic seizures
Cited by in corpus (4)
- Numerical Continuation and SPDE Stability for the 2D Cubic-Quintic Allen-Cahn Equation
- Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise
- Scaling Laws and Warning Signs for Bifurcations of SPDEs
- Critical slowing down associated with critical transition and risk of collapse in cryptocurrency