An elementary way to rigorously estimate convergence to equilibrium
and escape rates

Galatolo, Stefano; Nisoli, Isaia; Saussol, Benôıt

doi:10.3934/jcd.2015.2.51

Cited by 18 publications

(25 citation statements)

References 22 publications

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“…7 It can be difficult to find a sharp estimate for D. An approach allowing to find some useful upper estimates is shown in [17] 5.2. L ∞ norms.…”

Section: 1mentioning

confidence: 99%

Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

Galatolo¹,

Lucena²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation).As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size δ, the physical measure varies continuously, with a modulus of continuity O(δ log δ), which is asymptotically optimal for this kind of piecewise smooth maps. Acknowledgment This work was partially supported by Alagoas Research Foundation-FAPEAL (Brazil) Grants 60030 000587/2016, CNPq (Brazil) Grants 300398/2016-6, CAPES (Brazil) Grants 99999.014021/2013-07 and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS).2 Notation: In the following we use | · | to indicate the usual absolute value or norms for signed measures on the basis space N1. We will use || · || for norms defined for signed measures on Σ.

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“…7 It can be difficult to find a sharp estimate for D. An approach allowing to find some useful upper estimates is shown in [17] 5.2. L ∞ norms.…”

Section: 1mentioning

confidence: 99%

Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

Galatolo¹,

Lucena²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…Let m = 1/η, the space of zero average measures of a partition of size η has dimension m − 1; the way we compute (4.4) rigorously is to choose a basis of the space of average zero measures and explicitly multiply a sparse matrix with each element of this basis, which implies that the computation time scales asymptotically as m 2 (see [20,Section 8.3] for a complete treatment in the L 1 case). Therefore, to speed up computations, it is worth computing (4.4) on a coarser partition and get information on L by using [21].…”

Section: Estimating ||mentioning

confidence: 99%

A rigorous computational approach to linear response

Bahsoun

Galatolo

Nisoli³

et al. 2018

Nonlinearity

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We present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear response. We apply our results to expanding circle maps. In particular, we present examples where we compute, up to a pre-specified error in the L ∞norm, the response of expanding circle maps under stochastic and deterministic perturbations. Moreover, we present an example where we compute, up to a pre-specified error in the L 1 -norm, the response of the intermittent family at the boundary; i.e., when the unperturbed system is the doubling map.

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“…Since the calculus of L i δ,ξ | V is computationally complex, an alternative approach is used in [13] (see [22] for a previous application of a similar idea to deterministic dynamics). First, we use a coarser version of the operator, L δ contr ,ξ , where δ contr is a multiple of δ.…”

Section: Rigorous Approximationmentioning

confidence: 99%

Chaotic Itinerancy in Random Dynamical System Related to Associative Memory Models

et al. 2018

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We consider a random dynamical system arising as a model of the behavior of a macrovariable related to a more complicated model of associative memory. This system can be seen as a small (stochastic and deterministic) perturbation of a determinstic system having two weak attractors which are destroyed after the perturbation. We show, with a computer aided proof, that the system has a kind of chaotic itineracy. Typical orbits are globally chaotic, while they spend a relatively long time visiting the attractor's ruins.

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An elementary way to rigorously estimate convergence to equilibrium and escape rates

Cited by 18 publications

References 22 publications

Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

A rigorous computational approach to linear response

Chaotic Itinerancy in Random Dynamical System Related to Associative Memory Models

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