scholarly journals Non-autonomous invariant sets and attractors: Random dynamical system

2020 ◽  
Vol 25 (4) ◽  
pp. 17-23
Author(s):  
Mohamedsh Imran ◽  
Ihsan Jabbar Kadhim
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Global Attractors ◽  
Random Dynamical Systems ◽  
Invariant Sets ◽  
Random Dynamical System ◽  
Pullback Attractor ◽  
Pullback Attractors ◽  
Deterministic Dynamical System

 In this paper the concepts of pullback attractor ,pullback absorbing family in (deterministic) dynamical system are defined in (random) dynamical systems. Also some main result such as (existence) of pullback attractors ,upper semi-continuous of pullback attractors and uniform and global attractors are proved in random dynamical system .

scholarly journals Existence of random invariant periodic curves via random semiuniform ergodic theorem

2016 ◽  
Vol 17 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Kenneth Uda
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Dissipative Structure ◽  
Random Dynamical Systems ◽  
Random Dynamical System ◽  
Compact Set

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system on a cylinder [Formula: see text] has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

scholarly journals On the random dynamics of Volterra quadratic operators

2015 ◽  
Vol 37 (1) ◽  
pp. 228-243 ◽  
Author(s):  
U. U. JAMILOV ◽  
M. SCHEUTZOW ◽  
M. WILKE-BERENGUER
Keyword(s):  
Dynamical System ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Special Class ◽  
Random Dynamical Systems ◽  
Random Point ◽  
Random Dynamical System ◽  
Quadratic Stochastic Operators ◽  
Point Attractor ◽  
Set Up

We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex $S^{m-1}$. We prove that in contrast to the deterministic set-up the trajectories of the random dynamical system almost surely converge to one of the vertices of the simplex $S^{m-1}$, implying the survival of only one species. We also show that the minimal random point attractor of the system equals the set of all vertices. The convergence proof relies on a martingale-type limit theorem, which we prove in the appendix.

Structural stability of linear random dynamical systems

1996 ◽  
Vol 16 (6) ◽  
pp. 1207-1220 ◽  
Author(s):  
Nguyen Dinh Cong
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Structural Stability ◽  
Discrete Time ◽  
Random Dynamical Systems ◽  
Random Dynamical System ◽  
Time Linear ◽  
Structurally Stable

AbstractIn this paper, structural stability of discrete-time linear random dynamical systems is studied. A random dynamical system is called structurally stable with respect to a random norm if it is topologically conjugate to any random dynamical system which is sufficiently close to it in this norm. We prove that a discrete-time linear random dynamical system is structurally stable with respect to its Lyapunov norms if and only if it is hyperbolic.

scholarly journals Asymptotic Dynamics of Young Differential Equations

2021 ◽  
Author(s):  
Luu Hoang Duc ◽  
Phan Thanh Hong
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Random Dynamical Systems ◽  
Analytic Approach ◽  
Pullback Attractors ◽  
Random Attractors ◽  
Asymptotic Dynamics

AbstractWe provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

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.

A note on the generation of random dynamical systems from fractional stochastic delay differential equations

2015 ◽  
Vol 15 (03) ◽  
pp. 1550018 ◽  
Author(s):  
Luu Hoang Duc ◽  
Björn Schmalfuß ◽  
Stefan Siegmund
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Dynamical Systems ◽  
Delay Differential Equations ◽  
Regularity Conditions ◽  
Random Dynamical System ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Stochastic Delay Differential Equation ◽  
Delay Differential

In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Hölder space which is separable.

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