Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems
Maxime Breden, Maximilian Engel
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.
Topics & Concepts
MathematicsLyapunov exponentHopf bifurcationEigenfunctionConjectureBounded functionChaoticComputer-assisted proofComputationLyapunov functionPolynomial chaosMathematical analysisApplied mathematicsPure mathematicsEigenvalues and eigenvectorsBifurcationMathematical proofNonlinear systemStatisticsAlgorithmArtificial intelligenceComputer scienceMonte Carlo methodQuantum mechanicsPhysicsGeometryQuantum chaos and dynamical systemsMathematical Dynamics and FractalsNonlinear Dynamics and Pattern Formation