Equivalence of physical and SRB measures in random dynamical systems

Blumenthal, Alex; Young, Lai Sang

doi:10.1088/1361-6544/aafaa8

Cited by 8 publications

(6 citation statements)

References 22 publications

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“…The proof is a case-by-case verification of the above formula and is left to the reader. Recall that J OE "; " appearing in (13) has the property that points in z X mC1 C J have the same itinerary under Q…”

Section: Filtration H N /mentioning

confidence: 99%

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Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

Blumenthal

Yang

2022

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We study the effects of independent, identically distributed random perturbations of amplitude " > 0 on the asymptotic dynamics of one-parameter families ¹f a W S 1 ! S 1 ; a 2 OE0; 1º of smooth multimodal maps which are "predominantly expanding", i.e., jf 0 a j 1 away from small neighborhoods of the critical set ¹f 0 a D 0º. We obtain, for any " > 0, a checkable, finite-time criterion on the parameter a for random perturbations of the map f a to exhibit (i) a unique stationary measure and (ii) a positive Lyapunov exponent comparable to R S 1 log jf 0 a j dx. This stands in contrast with the situation for the deterministic dynamics of f a , the chaotic regimes of which are determined by typically uncheckable, infinite-time conditions. Moreover, our finite-time criterion depends on only k log." 1 / iterates of the deterministic dynamics of f a , which grows quite slowly as " ! 0.

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Section: Filtration H N /mentioning

confidence: 99%

“…For random systems with absolutely continuous stationary measures, a positivity of Lyapunov exponent implies existence of "random strange attractors" analogous to those for deterministic systems ([13,32]). …”

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confidence: 99%

Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

Blumenthal

Yang

2022

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…Numerical averages rely on ergodic convergences, and the stability of ergodic averages to noise is a complicated and broad question of its own. Stability results have been shown in systems with SRB measures W. Cowieson [2005], Blumenthal and Young [2019]. In such settings, the use of Kalman filtering in a model free approach Hamilton et al [2016Hamilton et al [ , 2017 may yield promising results.…”

Section: Future Workmentioning

confidence: 99%

Learning theory for dynamical systems

Berry¹,

Das²

2022

Preprint

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The task of modelling and forecasting a dynamical system is one of the oldest problems, and it remains challenging. Broadly, this task has two subtasks -extracting the full dynamical information from a partial observation; and then explicitly learning the dynamics from this information. We present a mathematical framework in which the dynamical information is represented in the form of an embedding. The framework combines the two subtasks using the language of spaces, maps, and commutations. The framework also unifies two of the most common learning paradigms -delaycoordinates and reservoir computing. We use this framework as a platform for two other investigations of the reconstructed system -its dynamical stability; and the growth of error under iterations. We show that these questions are deeply tied to more fundamental properties of the underlying systemthe behavior of matrix cocycles over the base dynamics, its non-uniform hyperbolic behavior, and its decay of correlations. Thus, our framework bridges the gap between universally observed behavior of dynamics modelling; and the spectral, differential and ergodic properties intrinsic to the dynamics.

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“…sample measures with densities on unstable manifolds. For details see [10,29] and for further discussions relevant for our setting e.g. [9,21].…”

Section: Chaotic Random Attractors and Singletonsmentioning

confidence: 99%

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

Engel

Kuehn

2021

Commun. Math. Phys.

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For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

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Equivalence of physical and SRB measures in random dynamical systems

Cited by 8 publications

References 22 publications

Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

Learning theory for dynamical systems

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

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