scholarly journals Mixing and observation for Markov operator cocycles*

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 66-83
Author(s):  
Fumihiko Nakamura ◽  
Yushi Nakano ◽  
Hisayoshi Toyokawa
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Markov Operator ◽  
Random Dynamical Systems ◽  
Asymptotically Periodic ◽  
Definition Of

Abstract We consider generalized definitions of mixing and exactness for random dynamical systems in terms of Markov operator cocycles. We first give six fundamental definitions of mixing for Markov operator cocycles in view of observations of the randomness in environments, and reduce them into two different groups. Secondly, we give the definition of exactness for Markov operator cocycles and show that Lin’s criterion for exactness can be naturally extended to the case of Markov operator cocycles. Finally, in the class of asymptotically periodic Markov operator cocycles, we show the Lasota–Mackey type equivalence between mixing, exactness and asymptotic stability.

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.

scholarly journals On the Computation of Invariant Measures in Random Dynamical Systems

2003 ◽  
Vol 03 (02) ◽  
pp. 247-265 ◽  
Author(s):  
Peter Imkeller ◽  
Peter Kloeden
Keyword(s):  
Dynamical Systems ◽  
Markov Chains ◽  
State Space ◽  
Difference Equations ◽  
Invariant Measures ◽  
Random Dynamical Systems ◽  
Continuum State ◽  
Transition Mechanism ◽  
Definition Of ◽  
Stationary Random Processes

Invariant measures of dynamical systems generated e.g. by difference equations can be computed by discretizing the originally continuum state space, and replacing the action of the generator by the transition mechanism of a Markov chain. In fact they are approximated by stationary vectors of these Markov chains. Here we extend this well-known approximation result and the underlying algorithm to the setting of random dynamical systems, i.e. dynamical systems on the skew product of a probability space carrying the underlying stationary stochasticity and the state space, a particular non-autonomous framework. The systems are generated by difference equations driven by stationary random processes modelled on a metric dynamical system. The approximation algorithm involves spatial discretizations and the definition of appropriate random Markov chains with stationary vectors converging to the random invariant measure of the system.

scholarly journals Random Dynamical Systems with Jumps and with a Function Type Intensity

2016 ◽  
Vol 30 (1) ◽  
pp. 63-87
Author(s):  
Joanna Kubieniec
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Invariant Measure ◽  
Lipschitz Function ◽  
Random Dynamical Systems ◽  
Function Type ◽  
Polish Spaces

AbstractIn paper [4] there are considered random dynamical systems with randomly chosen jumps acting on Polish spaces. The intensity of this process is a constant λ. In this paper we formulate criteria for the existence of an invariant measure and asymptotic stability for these systems in the case when λ is not constant but a Lipschitz function.

scholarly journals Strong Unique Ergodicity of Random Dynamical Systems on Polish Spaces

2016 ◽  
Vol 30 (1) ◽  
pp. 129-142
Author(s):  
Paweł Płonka
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Unique Ergodicity ◽  
Random Dynamical Systems ◽  
Random Measures ◽  
Polish Spaces

AbstractIn this paper we want to show the existence of a form of asymptotic stability of random dynamical systems in the sense of L. Arnold using arguments analogous to those presented by T. Szarek in [6], that is showing it using conditions generalizing the notion of tightness of measures. In order to do that we use tightness theory for random measures as developed by H. Crauel in [2].

VARIATIONAL PRINCIPLE FOR SUBADDITIVE SEQUENCE OF POTENTIALS IN BUNDLE RANDOM DYNAMICAL SYSTEMS

2009 ◽  
Vol 09 (02) ◽  
pp. 205-215 ◽  
Author(s):  
XIANFENG MA ◽  
ERCAI CHEN
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Multifractal Analysis ◽  
Weak Condition ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Set Up

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

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