RUELLE'S INEQUALITY FOR ISOTROPIC ORNSTEIN–UHLENBECK FLOWS

2010 ◽  
Vol 10 (01) ◽  
pp. 143-154 ◽  
Author(s):  
HOLGER VAN BARGEN
Keyword(s):  
Dynamical System ◽  
Random Dynamical System ◽  
Characteristic Numbers ◽  
Lyapunov Characteristic Numbers

Ruelle's inequality asserts that the entropy of a dynamical system is bounded from above by the Lyapunov characteristic numbers counted with their multiplicities. We show that this inequality holds true in the case of a random dynamical system deduced from an isotropic Ornstein–Uhlenbeck-flow (IOUF).

RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 369-388 ◽  
Author(s):  
M. J. GARRIDO-ATIENZA ◽  
A. OGROWSKY ◽  
B. SCHMALFUSS
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Random Dynamical System ◽  
Random Delay ◽  
Random Differential Equation ◽  
Autonomous Case ◽  
Random Differential Equations ◽  
Uniqueness Of A Solution

We investigate a random differential equation with random delay. First the non-autonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case.

Routes to Chaos in Neural Networks with Random Weights

1998 ◽  
Vol 08 (07) ◽  
pp. 1463-1478 ◽  
Author(s):  
D. J. Albers ◽  
J. C. Sprott ◽  
W. D. Dechert
Keyword(s):  
Dynamical System ◽  
Neural Networks ◽  
Function Space ◽  
Dynamic Properties ◽  
Monte Carlo Study ◽  
Random Dynamical System ◽  
Network Function ◽  
The Neural Network ◽  
Quasiperiodic Route To Chaos ◽  
Route To Chaos

Neural networks are dense in the space of dynamical system. We present a Monte Carlo study of the dynamic properties along the route to chaos over random dynamical system function space by randomly sampling the neural network function space. Our results show that as the dimension of the system (the number of dynamical variables) is increased, the probability of chaos approaches unity. We present theoretical and numerical results which show that as the dimension is increased, the quasiperiodic route to chaos is the dominant route. We also qualitatively analyze the dynamics along the route.

Random attractors for 3D Benjamin–Bona–Mahony equations derived by a Laplace-multiplier noise

2017 ◽  
Vol 18 (01) ◽  
pp. 1850004 ◽  
Author(s):  
Yangrong Li ◽  
Renhai Wang
Keyword(s):  
Dynamical System ◽  
Multiplicative Noise ◽  
Careful Analysis ◽  
Random Dynamical System ◽  
Stochastic Pde ◽  
Limit Set ◽  
Cutoff Function ◽  
Random System ◽  
Random Attractors ◽  
Bbm Equation

This paper contributes the dynamics for stochastic Benjamin–Bona–Mahony (BBM) equations on an unbounded 3D-channel with a multiplicative noise. An interesting feature is that the noise has a Laplace-operator multiplier, which seems not to appear in any literature for the study of stochastic PDE. After translating the stochastic BBM equation into a random equation and deducing a random dynamical system, we obtain both existence and semi-continuity of random attractors for this random system in the Sobolev space. The convergence of the system can be verified without the lower bound assumption of the nonlinear derivative. The tail-estimate is achieved by using a square of the usual cutoff function and by a careful analysis of the solution’s biquadrate. A spectrum method is also applied to prove the collective limit-set compactness.

scholarly journals Random dynamical system in time domain: A POD-PC model

2019 ◽  
Vol 133 ◽  
pp. 106251 ◽  
Author(s):  
E. Jacquelin ◽  
N. Baldanzini ◽  
B. Bhattacharyya ◽  
D. Brizard ◽  
M. Pierini
Keyword(s):  
Dynamical System ◽  
Time Domain ◽  
Random Dynamical System

Global random attractor of stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise on ℝd

2015 ◽  
Vol 16 (01) ◽  
pp. 1650007 ◽  
Author(s):  
Yanfeng Guo ◽  
Chunxiao Guo ◽  
Yongqian Han
Keyword(s):  
Dynamical System ◽  
Multiplicative Noise ◽  
Stochastic Equation ◽  
Random Coefficients ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Hydrodynamical Equation ◽  
Priori Estimates ◽  
Heisenberg Paramagnet ◽  
Truncation Function

The stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise defined on the entire [Formula: see text] is mainly investigated. The global random attractor for the random dynamical system associated with the equation is obtained. The method is to transform the stochastic equation into the corresponding partial differential equations with random coefficients by Ornstein–Uhlenbeck process. The uniform priori estimates for far-field values of solutions have been studied via a truncation function, and then the asymptotic compactness of the random dynamical system is established.

scholarly journals ASYMPTOTIC COMPACTNESS OF STOCHASTIC COMPLEX GINZBURG–LANDAU EQUATION ON AN UNBOUNDED DOMAIN

2010 ◽  
Vol 10 (04) ◽  
pp. 613-636 ◽  
Author(s):  
DIRK BLÖMKER ◽  
YONGQIAN HAN
Keyword(s):  
Dynamical System ◽  
Pattern Formation ◽  
White Noise ◽  
Unbounded Domain ◽  
Landau Equation ◽  
Random Dynamical System ◽  
Ginzburg Landau ◽  
Type Complex ◽  
Regularity Properties ◽  
Formation Systems

The Ginzburg–Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. In this paper, we consider the complex Ginzburg–Landau (CGL) equations on the whole real line perturbed by an additive spacetime white noise. Our main result shows that it generates an asymptotically compact stochastic or random dynamical system. This is a crucial property for the existence of a stochastic attractor for such CGL equations. We rely on suitable spaces with weights, due to the regularity properties of spacetime white noise, which gives rise to solutions that are unbounded in space.

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