scholarly journals Synchronization, Lyapunov Exponents and Stable Manifolds for Random Dynamical Systems

2018 ◽  
pp. 359-366 ◽  
Author(s):  
Michael Scheutzow ◽  
Isabell Vorkastner
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Random Dynamical Systems ◽  
Stable Manifolds

scholarly journals A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

2021 ◽  
Author(s):  
Maximilian Engel ◽  
Christian Kuehn
Keyword(s):  
Dynamical Systems ◽  
Limit Cycle ◽  
Level Sets ◽  
Return Time ◽  
Random Dynamical Systems ◽  
Expected Return ◽  
Stable Manifolds ◽  
Kolmogorov Operator ◽  
Definition Of ◽  
Stochastic Oscillations

AbstractFor 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.

Random entropy expansiveness for diffeomorphisms with dominated splittings

2019 ◽  
Vol 20 (02) ◽  
pp. 2050014
Author(s):  
Zeya Mi
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Random Dynamical Systems ◽  
Local Entropy ◽  
Dominated Splitting ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Bowen Balls ◽  
Partially Hyperbolic Diffeomorphisms

We study the local entropy of typical infinite Bowen balls in random dynamical systems, and show the random entropy expansiveness for [Formula: see text] partially hyperbolic diffeomorphisms with multi one-dimensional centers. Moreover, we consider [Formula: see text] diffeomorphism [Formula: see text] with dominated splitting [Formula: see text] such that [Formula: see text] for every [Formula: see text], and all the Lyapunov exponents are non-negative along [Formula: see text] and non-positive along [Formula: see text], we prove the asymptotically random entropy expansiveness for [Formula: see text].

scholarly journals ABSOLUTE CONTINUITY THEOREM FOR RANDOM DYNAMICAL SYSTEMS ON Rd

2013 ◽  
Vol 13 (02) ◽  
pp. 1250018 ◽  
Author(s):  
MORITZ BISKAMP
Keyword(s):  
Dynamical Systems ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Random Dynamical Systems ◽  
Random Dynamical System ◽  
Stable Manifolds ◽  
Continuity Theorem ◽  
Local Stable Manifold ◽  
Lebesgue Measures ◽  
Local Stable Manifolds

In this paper we provide a proof of the so-called absolute continuity theorem for random dynamical systems on Rd which have an invariant probability measure. First we present the construction of local stable manifolds in this case. Then the absolute continuity theorem basically states that for any two transversal manifolds to the family of local stable manifolds, the induced Lebesgue measures on these transversal manifolds are absolutely continuous under the map that transports every point on the first manifold along the local stable manifold to the second manifold, the so-called Poincaré map or holonomy map. In contrast to known results, we have to deal with the non-compactness of the state space and the randomness of the random dynamical system.

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