Exact potentials in multivariate Langevin equations

Abstract

Systems governed by a multivariate Langevin equation featuring an exact potential exhibit straightforward dynamics but are often difficult to recognize because, after a general coordinate change, the gradient flow becomes obscured by the Jacobian matrix of the mapping. In this work, a detailed analysis of the transformation properties of Langevin equations under general nonlinear mappings is presented. We show how to identify systems with exact potentials by understanding their differential-geometric properties. To demonstrate the power of our method, we use it to derive exact potentials for broadly studied models of nonlinear deterministic and stochastic oscillations. In selected examples, we visualize the identified potentials. Our results imply a broad class of exactly solvable stochastic models which can be self-consistently defined from given deterministic gradient systems.

arXiv comment9 pages, 5 figures

References in corpus (11)

• Noise Induced Complexity: From Subthreshold Oscillations to Spiking in Coupled Excitable Systems
• Diffusive dynamics and stochastic models of turbulent axisymmetric wakes
• Collective behaviour of swarmalators on a 1D ring
• Learning force fields from stochastic trajectories
• Reduced-Order Models for Coupled Dynamical Systems: Data-driven Methods and the Koopman Operator
• Symmetry breaking of azimuthal waves: Slow-flow dynamics on the Bloch Sphere
• Effective long-time phase dynamics of limit-cycle oscillators driven by weak colored noise
• Numerical Study of a Lyapunov Functional for the Complex Ginzburg-Landau Equation
• Effective Fokker-Planck Equation for Birhythmic Modified van der Pol Oscillator
• Stochastic modelling of a noise driven global instability in a turbulent swirling jet
• Averaging of thermoacoustic azimuthal instabilities