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Synchronization of noisy dissipative systems under discretization
Abstract
It is shown that the synchronization of noisy dissipative systems is preserved when a drift-implicit Euler scheme is used for the discretization. In particular, in this case the order of discretization and synchronization can be exchanged.
... There are very few papers dealing specifically with numerical stability of schemes applied to nonlinear SDEs, though there are some related papers that look at similar issues in the context of stochastic numerical dynamics, see Caraballo and Kloeden [8,9], Garrido-Atienza et al. [11], Kloeden et al. [20], Kloeden et al. [21]. These use concepts and methods from the theory of random dynamical systems. ...
... The main difference is that explicit formulas are not known for the corresponding stochastic stationary solution and their existence has to be established using the theory of random dynamical systems, see, e.g., Arnold [2]. The paper [9] discusses the implicit Euler-Maruyama scheme for the random dynamical equation to a Galerkin SDE with contractive nonlinearity obtained by Galerkin approximation of a SPDE with additive noise, thus an SDE of the form (3.4), whereas a second paper [21] discusses the implicit Euler-Maruyama scheme applied to a fully nonlinear Itô SDE with contractive one-sided Lipschitz continuous drift vector field f (3.2). The authors find that the numerical scheme has a stochastic stationary solution which attracts all other solutions pathwise in the pullback and the forward sense. ...
... for a constant time step size h > 0. 2) and the integrability condition (3.6). Then Theorem 3.1 states the same result as obtained in [21]. The fact that the implicit Euler method is the only θ-method which is stable for all equations satisfying a contractive one-sided Lipschitz condition mirrors precisely the B-stability result from deterministic numerics [32]. ...
An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.
... In [3], the author proved that the presence of additive noise could lead to a strengthening of the synchronization, i.e., the level of trajectories rather than attractors, which does not occur in the absence of noise. Moreover, the effects of discretization on the synchronization of dissipative systems with additive noise were investigated by Kloeden et al. [9], where it was seen that the synchronization effect was preserved using the drift-implicit Euler-Maruyama scheme with constant step size applied to the coupled system. ...
... then we transform (9) to the pathwise random ordinary differential equation ...
The synchronization of stochastic differential equations (SDEs) with additive noise is investigated in pathwise sense, moreover convergence rate of synchronization is obtained. The optimality of the convergence rate is illustrated through examples.
... This has been investigated mathematically by Afraimovich & Rodrigues [1], Carvalho et al. [6] and Rodrigues [19] for asymptotically stable equilibria and for general attractors, such as chaotic attractors. Analogous results also hold for nonautonomous systems [10] and noisy systems [4,14], but require new concepts of nonautonomous attractors and random attractors. The aim of this article is to introduce the reader to recent results on the dissipative synchronization of nonanutonomous and random dynamical systems as well as to new mathematical ideas and tools from the theories of nonautonomous and random dynamical systems needed to investigate them. ...
... The effects of discretization on the synchronization of dissipative systems with additive noise were investigated by Kloeden et al. [14], where it was seen that the synchronization effect was preserved using the drift-implicit Euler-Maruyama scheme [15] with constant step size Δt > 0 applied to the coupled system (16). ...
Recent results on the dissipative synchronization of nonanutonomous and random dynamical systems are discussed as well as new mathematical ideas and tools from the theories of nonautonomous and random dynamical systems that are needed for their formulation and investigation (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
... There are very few papers dealing specifically with the stability of numerical schemes for nonlinear SDEs driven by fBm. Some related papers look at a similar issue for nonlinear SDEs driven by Brownian motion; see [4,21]. These papers discuss the implicit Euler-Maruyama method and find that it has a unique stochastic stationary solution that attracts all other solutions pathwise in the pullback and the forward sense. ...
We discuss a system of stochastic differential equations with a stiff linear term and additive noise driven by fractional Brownian motions (fBms) with Hurst parameter H>1/2, which arise e. g., from spatial approximations of stochastic partial differential equations. For their numerical approximation, we present an exponential Euler scheme and show that it converges in the strong sense with an exact rate close to the Hurst parameter H. Further, based on [2], we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.
It is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise, no matter how large the intensity of the noise, provided asymptotically stable stationary-stochastic solutions are used instead of asymptotically stable equilibria.
In this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize. We apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems. The technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem, explicitly exhibited, in the diagonal.
The synchronization of two nonautonomous dynamical systems is considered, where the systems are described in terms of a skew-product formalism, i. e., in which an inputed autonomous driving system governs the evolution of the vector field of a differential equation with the passage of time. It is shown that the coupled trajectories converge to each other as time increases for sufficiently large coupling coefficient and also that the component sets of the pullback attractor of the coupled system converges upper semi continuously as the coupling parameter increases to the diagonal of the product of the corresponding component sets of the pullback attractor of a system generated by the average of the vector fields of the original uncoupled systems.
This chapter consists of a selection of examples from the literature of applications of stochastic differential equations. These are taken from a wide variety of disciplines with the aim of stimulating the readers’ interest to apply stochastic differential equations in their own particular fields of interest and of providing an indication of how others have used models described by stochastic differential equations. Here we simply describe the equations and refer readers to the original papers for the justification and analysis of the models.
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. Introduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.
The synchronization of Stratonovich stochastic differential equations (SDE) with a one-sided dissipative Lipschitz drift and linear multiplicative noise is investigated by transforming the SDE to random ordinary differential equations (RODE) and synchronizing their dynamics. In terms of the original SDE, this gives synchronization only when the driving noises are the same. Otherwise, the synchronization is modulo exponential factors involving Ornstein–Uhlenbeck processes corresponding to the driving noises. Moreover, this occurs no matter how large the intensity coefficients of the noise.
Switching systems formed by switching between a finite number of autonomous ordinary differential equations are formulated as abstract nonautonomous dynamical systems driven by the shift operator on the space of switching controls. A metric is proposed for such switching controls and the space of switching controls with a given positive dwell time is shown to be compact. This allows known results on pullback attractors for nonautonomous dynamical systems to be applied to switching systems including conditions which ensure that a pullback attractor exists and is also a forward attractor. In particular, this provides a conceptual framework for investigating the asymptotic behaviour of switching systems beyond the usual case in which the asymptotic stability of the zero solution under switching is considered.
The synchronization of two discrete time dynamical systems is considered, where the systems are described in terms of first order difference equations, each of which satisfies a global dissipativity condition and hence has a global attractor. It is shown that the coupled trajectories converge to each other as time increases for sufficiently large coupling coefficient and that the global attractor of the coupled system converges upper semi continuously as the coupling parameter increases to the diagonal of the product of the global attractor of a discrete time dynamical system for which the defining function is the average of those of the original uncoupled systems.
In this work we present some mathematical methods to obtain uniform estimates for attractors and to study the synchronization of two similar dynamical systems. We give sufficient conditions to control the coupling devices in order to accomplish the synchronization, even when each of the systems moves chaotically. Some examples that include systems of coupled lasers and coupled Lorenz equations are discussed.
We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f0 and the diffusion vector fields f1,...,fm, and investigate the existence of global random attractors for the associated flows . For this purpose is decomposed into a stationary diffeomorphism given by the stochastic differential equation on the space of smooth flows on Rd driven by m independent stationary Ornstein Uhlenbeck processes z1,...,zm and the vector fields f1,...,fm, and a flow generated by the nonautonomous ordinary differential equation given by the vector field (t/x)–1[f0(t)+
i=1
1
fi(t)z
t
i
]. In this setting, attractors of are canonically related with attractors of . For , the problem of existence of attractors is then considered as a perturbation problem. Conditions on the vector fields are derived under which a Lyapunov function for the deterministic differential equation determined by the vector field f0 is still a Lyapunov function for , yielding an attractor this way. The criterion is finally tested in various prominent examples.
Random dynamical systems are intrinsically nonautonomous and are formulated in terms of cocycles rather than semigroups. They
consequently require generalizations of the commonly used dynamical systems concepts such as attractors and invariance of
sets. This cocycle formalism is reviewed here and then the approximation of such dynamical behaviour by time discretized numerical
schemes is discussed, outlining results that have been obtained and those that remain to be resolved.
Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the
reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions.
A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations.
Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends
to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension
increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic
counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions.
. Random dynamical systems are intrinsically nonautonomous and are formulated in terms of cocycles rather than semigroups. They consequently require generalizations of the commonly used dynamical systems concepts such as attractors and invariance of sets. This formalism is briefly reviewed here and then the approximation of such dynamical behaviour by time discretized numerical schemes is discussed, outlining results that have been obtained and those that remain to be resolved. 1. Introduction The theory of random and stochastic dynamical systems, the foundations of which are expounded by Ludwig Arnold in the recent monograph [1], is a rich and profound synthesis of ideas, methods and results from ergodic theory and stochastic analysis with those from the theory of deterministic dynamical systems. During the developmental stages of this theory, which is still far from complete, numerical simulations of specific random and stochastic systems were used to obtain key insights into what c...
Synchronization of systems with multiplicative noise
- T Caraballo
- P E Kloeden
- A Neuenkirch
T. Caraballo, P.E. Kloeden, and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch.
Dyn. (2008), to appear.
Uniform dissipativeness and synchronization of nonautonomous equations
- V S Afraimovich
- H M Rodrigues
V.S. Afraimovich and H.M. Rodrigues, Uniform dissipativeness and synchronization of nonautonomous
equations, in International Conference on Differential Equations (Lisboa 1995), World Scientific
Publishing, River Edge, NJ, 1998, pp. 3 -17.