On common linear copositive Lyapunov functions for pairs of stable positive linear systems

We study the common linear copositive Lyapunov functions of positive linear systems. Firstly, we present a theorem on pairs of second order positive linear systems, and give another proof of this theorem by means of properties of geometry. Based on the process of the proof, we extended the results to a finite number of second order positive linear systems. Then we extend this result to third order systems. Finally, for higher order systems, we give some results on common linear copositive Lyapunov functions.     

Estimates and exact expressions for lyapunov exponents of stochastic linear differential equations

We consider the asymptotic behavior of the solutions of a stochastic linear differential equation driven by a finite states Markov process. We consider the sample path Lyapunov exponent λ and the p-moment Lyapunov exponents g(p) for positive p. We derive relations between X and g{p\ which are extensions to our situation of results of Arnold [1] in a different context. Using a Lyapunov function approach, an exact expression forg(2) and estimates for g(p) are obtained, thus leading to upper and lower bounds for λ     

Stability analysis of time-varying switched systems via indefinite difference Lyapunov functions

Asymptotic stability of time-varying switched systems is investigated in this paper. The less conservative sufficient criteria for asymptotic stability of time-varying discrete-time switched systems are proposed via common indefinite difference Lyapunov functions and multiple indefinite difference Lyapunov functions introduced in this note, respectively. Common indefinite difference Lyapunov functions can be used to analyze stability of a switched system with asymptotic stable subsystems and arbitrary switching signal. Multiple indefinite difference Lyapunov functions can be used to investigate stability of a switched system with unstable subsystems and a given switching signal. The difference of the proposed Lyapunov function may be positive at some instants for an asymptotically stable subsystem. We compare these main results and illustrate the effectiveness of the obtained theorems by three numerical examples.     

Constructions of strict Lyapunov functions for discrete time and hybrid time-varying systems

We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov’s Theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions.     

Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1

We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.     

Solution of Parabolic Equations by Means of Lyapunov Functionals

We propose a new approach to defining the notion of a solution to linear and nonlinear parabolic equations. The main idea consists in studying connections between solutions to dynamic problems in the variational shape and the properties of the corresponding Lyapunov functionals which are strictly decreasing along the trajectories of the above-mentioned dynamic equations except for the equilibrium points. It turns out that the families of Lyapunov functionals constructed by T. I. Zelenyak enable us to propose a new approach to defining solutions to both linear and nonlinear parabolic problems. All results are given in the case of smooth solutions. Note that the Lyapunov functionals can be used for studying solutions with unbounded gradients.     

An alternative proof of the Barker, Berman, Plemmons (BBP) result on diagonal stability and extensions

We revisit the theorem of Barker, Berman and Plemmons on the existence of a diagonal quadratic Lyapunov function for a stable linear time-invariant (LTI) dynamical system [G.P. Barker, A. Berman, R.J. Plemmons, Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra 5(3) (1978) 249-256]. We use recently derived results to provide an alternative proof of this result and to derive extensions.     

Lyapunov’s coefficient of regularity and nonuniform hyperbolicity

For a nonautonomous dynamics with discrete time defined by a sequence of matrices, we obtain sharp lower and upper bounds for the Lyapunov coefficient of regularity. This has the advantage of avoiding considering the adjoint dynamics (in contrast to what happens with the regularity coefficients considered by Perron and Grobman). We also show that the dynamics can always be reduced to one defined by upper triangular matrices with the additional properties that the canonical basis is normal and ordered. Moreover, we show in a simpler manner that the Lyapunov coefficient of regularity is related to the notion of nonuniform hyperbolicity, more precisely to the nonuniform part of a nonuniform exponential contraction or a nonuniform exponential dichotomy. Finally, as an application of this relation, we show that from the point of view of ergodic theory, for almost all trajectories with negative Lyapunov exponents the nonuniformity can be made arbitrarily small.     

Construction of Lyapunov functionals for delay differential equations in virology and epidemiology

In the present paper, we present a method for constructing a Lyapunov functional for some delay differential equations in virology and epidemiology. Here some delays are incorporated to the original ordinary differential equations, for which a Lyapunov function is already obtained. We present simple and clear explanation of our method using some models whose Lyapunov functionals are already obtained. Moreover, we present several new results for constructing Lyapunov functionals using our method.     

Surface billiards and nonvanishing Lyapunov exponents

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system. We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents. Some examples of such surfaces are given in this article.     

Lyapunov exponents of free operators

Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C.M. Newman's “triangle” law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform.     

带跳的非线性随机微分方程的Lyapunov指数的估计

本文研究了带跳的非线性随机微分方程Lyapunov指数的估计,在适当的条件下,确定其Lyapunov指数q的值．对于给定的步长h,考虑此微分系统的Euler离散化模型,给出了的理论误差估计.     

Stability of solutions to the Cauchy problem for a plane hyperbolic system with time-periodic coefficients

Considering a plane hyperbolic system with time-periodic coefficients, we construct a version of the direct Lyapunov method with the condition on the derivative of the Lyapunov functional along the trajectories of the system which is weakened by use of periodicity of the coefficients. We exhibit an illustrative example.     

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators

We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed as in [3], [4], while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.     

Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent

We give explicit examples of arbitrarily large analytic ergodic potentials for which the Schr?dinger equation has zero Lyapunov exponent for certain energies. For one of these energies there is an explicit solution. In the quasi-periodic case we prove that one can have positive Lyapunov exponent on certain regions of the spectrum and zero on other regions. We also show the existence of 1-dependent random potentials with zero Lyapunov exponent. Research partially supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), Institutional Grant 2002-2052. Received: February 2005; Accepted: May 2005     

Lyapunov pairs for perturbed sweeping processes

We give a full characterization of nonsmooth Lyapunov pairs for perturbed sweeping processes under very general hypotheses. As a consequence, we provide an existence result and a criterion for weak invariance for perturbed sweeping processes. Moreover, we characterize Lyapunov pairs for gradient complementarity dynamical systems.     

A coupling approach to estimating the Lyapunov exponent of stochastic max-plus linear systems

This paper addresses the problem of approximately computing the Lyapunov exponent of stochastic max-plus linear systems. Our approach allows for an efficient simulation of bounds for the Lyapunov exponent. We provide sufficient conditions for the convergence of the bounds. In particular, a perfect sampling scheme for the Lyapunov exponent is established. We illustrate the effectiveness of our bounds with an application to (real-life) railway systems.     

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. Given a suitable triangulation of a compact neighbourhood of the origin, a continuous and piecewise affine function can be parameterized by the values at the vertices of the triangulation. If these vertex values satisfy system-dependent linear inequalities, the parameterized function is a Lyapunov function for the system. We propose calculating these vertex values using constructions from two classical converse Lyapunov theorems originally due to Yoshizawa and Massera. Numerical examples are presented to illustrate the proposed approach.     

Lyapunov exponents for one-dimensional cellular automata

Summary In the paper we give a mathematical definition of the left and right Lyapunov exponents for a one-dimensional cellular automaton (CA). We establish an inequality between the Lyapunov exponents and entropies (spatial and temporal).