Random dynamical systems generated by coalescing stochastic flows on ℝ

2018 ◽  
Vol 18 (04) ◽  
pp. 1850031 ◽  
Author(s):  
Georgii V. Riabov
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
State Space ◽  
Random Dynamical Systems ◽  
Stochastic Flows

Existence of random dynamical systems for a class of coalescing stochastic flows on [Formula: see text] is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered.

scholarly journals Almost periodicity and periodicity for nonautonomous random dynamical systems

2020 ◽  
pp. 2150034
Author(s):  
Paul Raynaud de Fitte
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Random Processes ◽  
Random Dynamical Systems ◽  
Almost Periodicity ◽  
Almost Periodic

We present a notion of almost periodicity which can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept allows for improvements of known results of almost periodicity in distribution, for general random processes and for solutions to stochastic differential equations.

THE COHOMOLOGY OF STOCHASTIC AND RANDOM DIFFERENTIAL EQUATIONS, AND LOCAL LINEARIZATION OF STOCHASTIC FLOWS

2002 ◽  
Vol 02 (02) ◽  
pp. 131-159 ◽  
Author(s):  
PETER IMKELLER ◽  
CHRISTIAN LEDERER
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Calculus ◽  
Random Dynamical Systems ◽  
Stochastic Flows ◽  
Local Linearization ◽  
Random Differential Equations ◽  
Classical Analogue ◽  
Asymptotic Problems ◽  
The One

Random dynamical systems can be generated by stochastic differential equations (sde) on the one hand, and by random differential equations (rde), i.e. randomly parametrized ordinary differential equations on the other hand. Due to conflicting concepts in stochastic calculus and ergodic theory, asymptotic problems for systems associated with sde are harder to treat. We show that both objects are basically identical, modulo a stationary coordinate change (cohomology) on the state space. This observation opens completely new opportunities for the treatment of asymptotic problems for systems related to sde: just study them for the conjugate rde, which is often possible by simple path-by-path classical arguments. This is exemplified for the problem of local linearization of random dynamical systems, the classical analogue of which leads to the Hartman–Grobman theorem.

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

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.

Sign in / Sign up

Export Citation Format

Share Document