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

Breden, Maxime; Engel, Maximilian

doi:10.48550/arxiv.2101.01491

Cited by 4 publications

(5 citation statements)

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“…Since ν > 1 the associated function u is smooth, and is therefore a strong solution of (5). The announced error estimate in C 0 norm follows directly from (6).…”

Section: Example and Resultsmentioning

confidence: 93%

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Computer-assisted proofs for some nonlinear diffusion problems

Breden

2021

Preprint

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In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form ∆u+f (u, ∇u) = 0. By studying a well chosen fixed point problem defined around the numerical solution, these techniques make it possible to prove the existence of a solution in an explicit (and usually small) neighborhood the numerical solution. In this work, we develop a similar approach for a broader class of systems, including nonlinear diffusion terms of the form ∆Φ(u). In particular, this enables us to obtain new results about steady states of a cross-diffusion system from population dynamics: the (non-triangular) SKT model. We also revisit the idea of automatic differentiation in the context of computer-assisted proof, and propose an alternative approach based on differential-algebraic equations.

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“…Since ν > 1 the associated function u is smooth, and is therefore a strong solution of (5). The announced error estimate in C 0 norm follows directly from (6).…”

Section: Example and Resultsmentioning

confidence: 93%

“…However, in all these studies, the leading differential operator is always simply ∆u, or ∆ 2 u. The only exceptions seem to be [5] and [6], where some very specific non-constant coefficients in front of the Laplacian are considered.…”

Section: Computer-assisted Proofs For Elliptic Equationsmentioning

confidence: 99%

Computer-assisted proofs for some nonlinear diffusion problems

Breden

2021

Preprint

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“…They have transferred the notion of a dominant Lyapunov exponent to the setting of conditioning a stochastic system to remain within a bounded subdomain of the state space. Recently, this quantity has been used to prove a long conjectured bifurcation result from noise-induced synchronisation to chaos via rigorous numerics [9]. The main result of our work is an extension of this one averaged exponent to an entire Lyapunov spectrum for general RDS in the conditioned setting.…”

Section: Introductionmentioning

confidence: 92%

The Lyapunov spectrum for conditioned random dynamical systems

Castro¹,

Chemnitz²,

Chu³

et al. 2022

Preprint

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We establish the existence of a full spectrum of Lyapunov exponents for memoryless random dynamical systems with absorption. To this end, we crucially embed the process conditioned to never being absorbed, the Q-process, into the framework of random dynamical systems, allowing us to study multiplicative ergodic properties. We show that the finite-time Lyapunov exponents converge in conditioned probability and apply our results to iterated function systems and stochastic differential equations.

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“…λ 1 < 0, has been proven analytically [85]. The change of sign of λ 1 to positive values is only proven in the particular context of the conditioned Lyapunov exponent [108], considering the random dynamics on a bounded domain with killing at the boundary, by conducting a computer-assisted proof [109]. An explicit formula as before seems out of scope for system (X sH ) on the whole domain.…”

Section: Shear-induced Chaosmentioning

confidence: 99%