Linear Random Dynamical Systems Research Articles

Generalize of some concepts on Linear Random Dynamical Systems

This work aims to study the concepts of proximal, distal, minimal, regionally proximal, and uniformly equicontinuous in the linear random dynamical systems with the study of some their properties.

Bohl–Perron theorem for random dynamical systems

In this paper, we consider the Bohl–Perron Theorem for linear random dynamical systems. We prove that the tempered exponential stability of a linear co-cycle is equivalent to the boundedness of solutions for inherit difference equation. Paper also proves a similar concept for co-cycle admitting a tempered exponential dichotomy.

Stability analysis of semilinear stochastic differential equations

This paper is concerned with the global stability of semilinear stochastic differential equations (SDEs) with multiplicative white noise, which is a continuation of our recent work published in SIAM Journal on Control and Optimization, 2018. Under an explicit condition that the Lipschitz constant of nonlinear term is smaller than the top Lyapunov exponent of the linear random dynamical system (RDS), we prove that the zero solution is globally stable.

On Sensitivity and Expansivity of Linear Random Dynamical Systems

in this study we introduce and study of on Sensitivity and expansivity of Linear Random DYnamical systems
 We obtain various important properties and theorems about Sensitivity Random DYnamical systems and Expansivity Random DYnamical systems

Stability index of linear random dynamical systems

Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of the stable manifold of the zero solution. In particular, for the $n$ dimensional case, the zero solution is globally asymptotically stable if and only if this stability index is $n.$ Fixed $n,$ let $X$ be the random variable that assigns to each linear random dynamical system its stability index, and let $p_k$ with $k=0,1,\ldots,n,$ denote the probabilities that $P(X=k)$. In this paper we obtain either the exact values $p_k,$ or their estimations by combining the Monte Carlo method with a least square approach that uses some affine relations among the values $p_k,k=0,1,\ldots,n.$ The particular case of $n$-order homogeneous linear random differential or difference equations is also studied in detail.

Invariant Manifolds for Random Dynamical Systems on Banach Spaces Exhibiting Generalized Dichotomies

We prove the existence of measurable invariant manifolds for small perturbations of linear Random Dynamical Systems evolving on a Banach space and admitting a general type of dichotomy, both for continuous and discrete time. Moreover, the asymptotic behavior in the invariant manifold is similar to the one of the linear Random Dynamical System.

Reduced order modeling of random linear dynamical systems based on a new a posteriori error bound

SummaryReduced order models (ROMs) are becoming increasingly useful for saving computational cost in response prediction of vibrating systems. In a number of applications such as uncertainty quantification, ROMs require robustness over a wide variation of parameters. Accordingly, often they are classified as local and global, based on their performance in the parametric domain. Availability of an error bound of a ROM helps in achieving this robustness, mainly by allowing adaptivity. In this work, for a linear random dynamical system, first, an a posteriori error bound is developed based on the residual in the governing differential equation. Next, based on this error bound, two adaptive methods are proposed for building robust ROMs, that is, one for local, and another for global. While both methods are based on a greedy search approach, a modification is proposed in the training stage of the global ROM for accelerated convergence. These methods are then applied to an uncertainty quantification problem in a statistical simulation framework, and accordingly, two algorithms are developed. A detailed numerical study on vibration of a bladed disk assembly is conducted to study the accuracy and efficiency of the proposed error bound and adaptive ROMs. It is found that these adaptive ROMs provide a considerable speed‐up in estimating the probability of failure.

Balanced model order reduction for linear random dynamical systems driven by Lévy noise

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Levy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.

On random linear dynamical systems in a Banach space. I. Multiplicative Ergodic Theorem and Krein–Rutman type Theorems

For linear random dynamical systems in a separable Banach space X, we derived a series of Krein–Rutman type Theorems with respect to co-invariant cone family with rank-k, which present a (quasi)-equivalence relation between the measurably co-invariant cone family and the measurably dominated splitting of X. Moreover, such (quasi)-equivalence relation turns out to be an equivalence relation whenever (i) k=1; or (ii) in the frame of the Multiplicative Ergodic Theorem with certain Lyapunov exponent being greater than the negative infinity. For the second case, we thoroughly investigated the relations between the Lyapunov exponents, the co-invariant cone family and the measurably dominated splitting for linear random dynamical systems in X.

On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems

In this paper, we construct an open and dense set in the space of bounded linear random dynamical systems (both discrete and continuous time) equipped with the essential sup norm such that the Lyapunov exponents depend analytically on the coefficients in this set. As a consequence, analyticity for Lyapunov exponents of bounded linear random dynamical systems is a generic property.

K-dimensional invariant cones of random dynamical systems in [formula omitted] with applications

We investigated a linear random dynamical system which strongly preserves a cone C of dimension-k (abbr. k-cone) in Rn. Under some general assumptions, it is shown that such system admits a measurable family of k-dimensional subspaces and a measurable family of (n−k)-dimensional subspaces which are complementary to each other and form into a tempered invariant splitting of Rn. We further apply the measurable bundles so obtained to study the linear random monotone cyclic feedback systems, as well as the linear competitive–cooperative tridiagonal systems. This generalizes the Floquet theory for these deterministic non-autonomous (or time-periodic) systems to the random systems.

The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.

Roughness of tempered exponential dichotomies for infinite-dimensional random difference equations

In this paper we study the roughness of tempered exponential dichotomies for linear random dynamical systems in Banach spaces. Such a dichotomy has a tempered bound and describes nonuniform hyperbolicity. We prove the roughness without assuming their invertibility and the integrability condition of the Multiplicative Ergodic Theorem. We give an explicit bound for the linear perturbation such that the dichotomy is persistent. We also obtain explicit forms for the exponent and the bound of tempered exponential dichotomy of the perturbed random system in terms of the original ones and the perturbations.

Lyapunov regularity for random dynamical systems

This paper is concerned with the Lyapunov regularity of linear random dynamical systems. Both continuous and discrete cases are treated and the conditions are expressed in terms of the so-called regularity coefficient, which is an essential element of the theory of Lyapunov forward regularity developed by Lyapunov himself. It turns out that, using the celebrated multiplicative ergodic theorem of Oseledets, the Lyapunov forward regularity is “typical” in a measure theoretical sense.

ON THE ABSOLUTE REGULARITY OF LINEAR RANDOM DYNAMICAL SYSTEMS

We introduce a concept of absolute regularity of linear random dynamical systems (RDS) that is stronger than Lyapunov regularity. We prove that a linear RDS that satisfies the integrability conditions of the multiplicative ergodic theorem of Oseledets is not merely Lyapunov regular but absolutely regular.

Structural stability of linear random dynamical systems

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.

An asymptotically exact stopping rule for the numerical computation of the Lyapunov spectrum

It is in general not possible to analytically compute the Lyapunov spectrum of a given dynamical system. This has been achieved for a few special cases only. Therefore, numerical algorithms have been devised for this task. However, one major drawback of these numerical algorithms is their lack of stopping rules. In this paper, an asymptotically exact stopping rule is proposed to alleviate this shortcoming while computing the Lyapunov spectrum of linear discretetime random dynamical systems (i.e., linear systems with random parameters). The proposed stopping rule provides an estimate of the least number of iterations, for which the probability of incurring a prescribed error, in the numerical computation of the Lyapunov spectrum, is minimized. It exploits simple upper bounds on the Lyapunov exponents, along with some results from finite state Markov chains. The accuracy of the stopping rule, and the computational load, is proportional to the tightness of the bound. In fact, a series of increasingly tighter bounds are proposed, yielding an asymptotically exact stopping rule for the tightest one. It is demonstrated via an example, that the proposed stopping rule is applicable to nonlinear dynamics as well.