Probability, statistics and computation in dynamical systems

Galatolo, Stefano; Nisoli, Isaia; Rojas, Cristóbal

doi:10.1017/s0960129512000758

Cited by 5 publications

(3 citation statements)

References 33 publications

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“…The estimate given at (16) mainly depend on the ratio δξ −1 between the partition size and the noise amplitude. This estimate is obtained without any information on the deterministic part of the dynamics, only the information about the contraction rate of the approximated transfer operator L δ,ξ (to obtain n and α).…”

Section: A Bound Formentioning

confidence: 99%

“…It is known that naive computer simulations of chaotic systems may not be reliable in some case (see [8,[16][17][18]20] for examples of misleading naive simulations and a general discussion on the problem). Beside the pure mathematical interest of a rigorously proved result and a rigorously certi ed estimate, the study of inherently reliable methods for the numerical study of chaotic dynamical systems is strongly motivated.…”

Section: Introductionmentioning

confidence: 99%

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Existence of noise induced order, a computer aided proof

Galatolo

Monge²,

Nisoli³

2020

Nonlinearity

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We prove the existence of noise induced order in the Matsumoto-Tsuda model, where it was originally discovered in 1983 by numerical simulations. This is a model of the famous Belousov-Zhabotinsky reaction, a chaotic chemical reaction, and consists of a one dimensional random dynamical system with additive noise. The simulations showed that an increase in amplitude of the noise causes the Lyapunov exponent to decrease from positive to negative; we give a mathematical proof of the existence of this transition. The method we use relies on some computer aided estimates providing a certi ed approximation of the system's stationary measure in the L 1 norm. This is realized by explicit functional analytic estimates working together with an ef cient algorithm. The method is general enough to be adapted to any piecewise differentiable dynamical system on the unit interval with additive noise. We also prove that the stationary measure varies in a Lipschitz way if the system is perturbed and that the Lyapunov exponent of the system varies in a Hölder way when the noise amplitude increases.

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Section: A Bound Formentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Existence of noise induced order, a computer aided proof

Galatolo

Monge²,

Nisoli³

2020

Nonlinearity

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is known that naive computer simulations of chaotic systems may not be reliable in some case (see for example [7], [17], [15], [19], [16] for examples of misleading naive simulations and a general discussion on the problem). Beside the pure mathematical interest of a rigorously proved result and a rigorously certified estimate, the study of inherently reliable methods for the numerical study of chaotic dynamical systems is strongly motivated.…”

Section: Introductionmentioning

confidence: 99%

Existence of Noise Induced Order, a Computer Aided Proof

Galatolo¹,

Monge²,

Nisoli³

2017

Preprint

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We prove the existence of Noise Induced Order in the Matsumoto-Tsuda model, where it was originally discovered in 1983 by numerical simulations. This is a model of the famous Belosouv-Zabotinsky reaction, a chaotic chemical reaction, and consists of a one dimensional random dynamical system with additive noise. The simulations showed that an increase in amplitude of the noise causes the Lyapunov exponent to decrease from positive to negative; we give a mathematical proof of the existence of this transition. The method we use relies on some computer aided estimates providing a certified approximation of the stationary measure in the L 1 norm. This is realized by explicit functional analytic estimates working together with an efficient algorithm. The method is general enough to be adapted to any piecewise differentiable dynamical system on the unit interval with additive noise. We also prove that the stationary measure of the system varies in a Lipschitz way if the system is perturbed and that the Lyapunov exponent of the system varies in a Hölder way when the noise amplitude increases.

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Cramér distance and discretizations of circle expanding maps II: simulations

Guihéneuf

Monge²

2023

Dynamical Systems

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Probability, statistics and computation in dynamical systems

Cited by 5 publications

References 33 publications

Existence of noise induced order, a computer aided proof

Existence of noise induced order, a computer aided proof

Existence of Noise Induced Order, a Computer Aided Proof

Cramér distance and discretizations of circle expanding maps II: simulations

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