Continuity properties of transport coefficients in simple maps

Keller, Gerhard; Howard, P J; Klages, Rainer

doi:10.1088/0951-7715/21/8/003

Cited by 14 publications

(23 citation statements)

References 43 publications

(195 reference statements)

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“…A related comment we would like to make concerns Theorem 2 which proves that the invariant density depends on ε Lipschitz continuously. We think that this result is remarkable as in the presence of singularities typically only a weaker, log-Lipschitz continuous dependence can be expected ( [2], [4], [14]).…”

Section: Introductionmentioning

confidence: 77%

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

et al. 2018

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We study a class of globally coupled maps in the continuum limit, where the individual maps are expanding maps of the circle. The circle maps in question are such that the uncoupled system admits a unique absolutely continuous invariant measure (acim), which is furthermore mixing. Interaction arises in the form of diffusive coupling, which involves a function that is discontinuous on the circle. We show that for sufficiently small coupling strength the coupled map system admits a unique absolutely continuous invariant distribution, which depends on the coupling strength ε. Furthermore, the invariant density exponentially attracts all initial distributions considered in our framework. We also show that the dependence of the invariant density on the coupling strength ε is Lipschitz continuous in the BV norm.When the coupling is sufficiently strong, the limit behavior of the system is more complex. We prove that a wide class of initial measures approach a point mass with support moving chaotically on the circle. This can be interpreted as synchronization in a chaotic state.

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Section: Introductionmentioning

confidence: 77%

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

et al. 2018

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“…Indeed, since c is not a periodic point of f t ′ , there is ǫ 3 (t ′ ) > 0 such that (11) η := min {|f…”

Section: Good Transversal Familiesmentioning

confidence: 99%

Central Limit Theorem for the Modulus of Continuity of Averages of Observables on Transversal Families of Piecewise Expanding Unimodal Maps

Lima¹,

Smania²

2016

J. Inst. Math. Jussieu

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“…Remark 4.2. Item 2 of Corollary 4.1 has been proven before for perturbations of type (DP) and (NP) in [29] and [4], respectively.…”

Section: Theorem 34: the Approximate Variance λmentioning

confidence: 74%

Stability and Approximation of Statistical Limit Laws for Multidimensional Piecewise Expanding Maps

Crimmins

Froyland

2019

Ann. Henri Poincaré

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The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. In the abstract setting of Keller-Liverani [30] we prove that derivatives of all orders of the leading eigenvalues and eigenprojections of the twisted transfer operators with respect to the twist parameter are stable when subjected to a broad class of perturbations. As a result, we demonstrate stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We apply these results to piecewise expanding maps in one and multiple dimensions, including new convergence results for Ulam projections on quasi-Hölder spaces.

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Continuity properties of transport coefficients in simple maps

Cited by 14 publications

References 43 publications

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

Central Limit Theorem for the Modulus of Continuity of Averages of Observables on Transversal Families of Piecewise Expanding Unimodal Maps

Stability and Approximation of Statistical Limit Laws for Multidimensional Piecewise Expanding Maps

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