Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

Abstract

We confirm a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. The method of showing the main chaotic property, a positive Lyapunov exponent, is a computer-assisted proof. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains and the modified Furstenberg-Khasminskii formula, the problem boils down to the rigorous computation of eigenfunctions of the Kolmogorov operators describing distributions of the underlying stochastic process.

References in corpus (7)

• CAPD::DynSys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems
• Shear-Induced Chaos
• Lyapunov exponents for random perturbations of some area-preserving maps including the standard map
• Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems
• Numerical verification method for positive solutions of elliptic problems
• Lower bounds on the Lyapunov exponents of stochastic differential equations
• A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

Cited by in corpus (6)

• Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems
• Smooth imploding solutions for 3D compressible fluids
• Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise
• Computer-assisted proofs for some nonlinear diffusion problems
• A General View on Double Limits in Differential Equations
• The conditioned Lyapunov spectrum for random dynamical systems