Lyapunov Regularity Research Articles

Stability of non-instantaneous impulsive systems in Hilbert spaces

In this paper, we introduce the concepts of Lyapunov exponent and Lyapunov regularity in Hilbert spaces, which is similar to that in the classical theory. As an application, under sufficiently small perturbations, we establish the asymptotic stability of the zero solution of non-instantaneous impulses nonautonomous nonlinear perturbations systems, which linear system is Lyapunov regular.

Some applications of Lyapunov regularity

Some applications of Lyapunov regularity

New regularity coefficients

We give two new characterizations of the notion of Lyapunov regularity in terms of the lower and upper exponential growth rates of the singular values. These characterizations motivate the introduction of new regularity coefficients. In particular, we establish relations between these regularity coefficients and the Lyapunov regularity coefficient. Moreover, we construct explicitly bounded sequences of matrices attaining specific values of the new regularity coefficients.

Regularity Coefficients for Impulsive Differential Equations

Abstract For a linear impulsive differential equation, we introduce a Lyapunov regularity coefficient following as far as possible the non-impulsive case. We recall that a regularity coefficient is a quantity that characterizes the Lyapunov regularity of the dynamics. In particular, we obtain lower and upper bounds for the Lyapunov regularity coefficient and we show that its computation can always be reduced to that of the corresponding coefficient of an impulsive dynamics defined by upper triangular matrices. We also relate the Lyapunov regularity coefficient with the Grobman regularity coefficient. Finally, we combine all the former results to establish a criterion for tempered exponential behavior in terms of the Lyapunov exponents and of the Lyapunov regularity coefficient.

Regularity and Stability Sets for Families of Sequences of Matrices

We consider the notions of Lyapunov regularity and of Lyapunov stability and asymptotic stability for a dynamics defined by a continuous 1-parameter family of sequences of matrices. In particular, we identify all classes of sets that can be the regularity set, the stability set and the asymptotic stability set of any such family. Moreover, we construct explicitly families of sequences of matrices whose regularity set, stability set or asymptotic stability set is a given set in those classes.

Lyapunov regularity and stability of linear non-instantaneous impulsive differential systems

Abstract In this paper, we discuss Lyapunov regularity and stability for linear non-instantaneous impulsive differential systems. In particular, we give sufficient conditions to guarantee any non-trivial solution has a finite Lyapunov exponent and we prove an impulsive system is stable using the Lyapunov exponent for the solution. A new version of Perron’s theorem is given by introducing the associated adjoint impulsive system and some criteria for the existence of non-uniform exponential behaviour are given. In addition, we present a stability result for a small perturbed nonlinear impulsive system when the linear impulsive system admits a non-uniform exponential contraction. Finally, we give a bound for the regularity coefficient.

Transitively-saturated property, Banach recurrence and Lyapunov regularity

The topological entropy of upper recurrent, lower recurrent points and Birkhoff regularity were considered in Tian (2016 Adv. Math. 288 464–526) but Banach upper recurrence points are not considered. In this paper we mainly consider whether the points that are Banach upper recurrent but not upper recurrent can carry full topological entropy. Moreover, we use statistical -limit sets to obtain entropy results on the refined orbit distribution of Banach recurrence. In this process, we strengthen Pfister and Sullivan’s result of Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst. 27 929–56) from saturated property to transitively-saturated property (and from single-saturated property to transitively-convex-saturated property). We also establish an abstract framework on the combination of Banach recurrence and multifractal analysis of general observable functions and in particular, how these results can be applied in the combination of Banach recurrence Lyapunov regularity to obtain a generalized and refined theory of Barreira and Doutor (2009 J. Math. Pures Appl. 92 1–17); Barreira and Gelfert (2006 Commun. Math. Phys. 267 393–418); Barreira and Saussol (2001 Trans. Am. Math. Soc. 353 3919–44); Feng and Huang (2010 Commun. Math. Phys. 297 1–43); Feng (2003 J. Math. 138 353–76); Feng (2009 Israel J. Math. 170 355–94); Feng and Lau (2002 Math. Res. Lett. 9 363–78) etc, on mixed Lyapunov multifractal analysis such as irregular sets and level sets of Lyapunov exponents.

Proportional Local Assignability of Lyapunov Spectrum of Linear Discrete Time-Varying Systems

We consider a local version of the pole assignment problem for linear discrete time-varying systems. Our aim is to obtain sufficient conditions to place the Lyapunov spectrum of the closed-loop system in an arbitrary position within some neighborhood of the Lyapunov spectrum of the free system using an appropriate time-varying linear feedback. Moreover, we assume that the norm of the matrix of linear feedback should be bounded from above by the distance between these two spectra with some constant multiplier. We prove that diagonalizability, Lyapunov regularity, and stability of the Lyapunov spectrum each separately are the required sufficient conditions provided that the open-loop system is uniformly completely controllable.

Lyapunov regularity and triangularization for unbounded sequences

Lyapunov regularity and triangularization for unbounded sequences

Dichotomy spectrum for difference equations in Banach spaces

In this paper, we study the structure and properties of the dichotomy spectrum for linear difference equations in Banach spaces avoiding the commonly made assumption of having a compact flow. In this situation, any compact subset of the positive-half line may occur as a dichotomy spectrum. To overcome this difficulty, we introduce the upper and lower dichotomy index by means of a measure of noncompactness. It allows to decompose the dichotomy spectrum into the upper, essential and lower dichotomy spectrum. The main result is a Spectral Theorem which describes all possible cases of the upper and lower dichotomy spectrum and yields a “nonautonomous linear algebra” appropriate also for systems in the absence of Lyapunov regularity. Furthermore, we give explicit examples of difference equations to illustrate the different cases of the Spectral Theorem and to underline that a new form of the spectrum not present in the literature can occur. Finally, we indicate applications to nonlinear difference equations and the continuous time situation.

New answers to an old question in the theory of differential–algebraic equations: Essential underlying ODE versus inherent ODE

In the context of linear differential–algebraic equations (DAEs) one finds different associated explicit ordinary differential equations (ODEs), among them essential underlying and inherent explicit regular ones, abbreviated: EUODEs and IERODEs. EUODEs have been introduced in 1991 for index-2 DAEs in Hessenberg form by means of special transformations. IERODEs result within the framework of the projector based decoupling. Each such explicit ODE is occasionally considered to rule the flow of the DAE.The question to which extend EUODEs and IERODEs are related to each other has been asked promptly after 1991. For index-2 Hessenberg-form DAEs, answers have been given in 2005, saying that EUODEs represent somehow condensed IERODEs. Recently, EUODEs have been indicated for general arbitrary-index DAEs and it has been proved that they are condensed IERODEs. The understanding of the relation between the IERODE and the EUODEs enables to uncover the stability behavior of the DAE flow.In the present paper we show that both, the IERODEs and EUODEs of a DAE with arbitrary high index do not at all depend on derivatives of the right-hand side. We consider adjoint pairs of DAEs and provide generalizations of the classical Lagrange identity. Furthermore, we address Lyapunov spectra and Lyapunov regularity.

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.

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.

Lyapunov regularity of impulsive differential equations

For linear impulsive differential equations, we give a simple criterion for the existence of a nonuniform exponential dichotomy, which includes uniform exponential dichotomies as a very special case. For this we introduce the notion of Lyapunov regularity for a linear impulsive differential equation, in terms of the so-called regularity coefficient. The theory is then used to show that if the Lyapunov exponents are nonzero, then there is a nonuniform exponential behavior, which can be expressed in terms of the Lyapunov exponents of the differential equation and of the regularity coefficient. We also consider the particular case of nonuniform exponential contractions when there are only negative Lyapunov exponents. Having this relation in mind, it is also of interest to provide alternative characterizations of Lyapunov regularity, and particularly to obtain sharp lower and upper bound for the regularity coefficient. In particular, we obtain bounds expressed in terms of the matrices defining the impulsive linear system, and we obtain characterizations in terms of the exponential growth rate of volumes. In addition we establish the persistence of the stability of a linear impulsive differential equation under sufficiently small nonlinear perturbations.

Polynomial contractions and Lyapunov regularity

Abstract The purpose of this note is twofold: to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time, and to show that it persists under sufficiently small linear and nonlinear perturbations. The notion of polynomial contraction mimics the notion of exponential contraction, but with the exponential decay replaced by a polynomial decay. We show that this behavior is exhibited by a large class of dynamics, by giving necessary conditions in terms of “polynomial” Lyapunov exponents. Finally, we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations. We also consider the case of nonuniform polynomial contractions, for which the Lyapunov stability is not uniform.

Stability theory and Lyapunov regularity

We establish the stability under perturbations of the dynamics defined by a sequence of linear maps that may exhibit both nonuniform exponential contraction and expansion. This means that the constants determining the exponential behavior may increase exponentially as time approaches infinity. In particular, we establish the stability under perturbations of a nonuniform exponential contraction under appropriate conditions that are much more general than uniform asymptotic stability. The conditions are expressed in terms of the so-called regularity coefficient, which is an essential element of the theory of Lyapunov regularity developed by Lyapunov himself. We also obtain sharp lower and upper bounds for the regularity coefficient, thus allowing the application of our results to many concrete dynamics. It turns out that, using the theory of Lyapunov regularity, we can show that the nonuniform exponential behavior is ubiquitous, contrarily to what happens with the uniform exponential behavior that although robust is much less common. We also consider the case of infinite-dimensional systems.

Energy–second-moment map analysis as an approach to quantify the irregularity of Hamiltonian systems

A different approach will be presented that aims to scrutinize the phase-space trajectories of a general class of Hamiltonian systems with regard to their regular or irregular behavior. The approach is based on the "energy-second-moment map" that can be constructed for all Hamiltonian systems of the generic form H=p(2)/2+V(q,t) . With a three-component vector s consisting of the system's energy h and second moments qp, q(2), this map linearly relates the vector s(t) at time t with the vector's initial state s(0) at t=0 . It will turn out that this map is directly obtained from the solution of a linear third-order equation that establishes an extension of the set of canonical equations. The Lyapunov functions of the energy-second-moment map will be shown to have simple analytical representations in terms of the solutions of this linear third-order equation. Applying Lyapunov's regularity analysis for linear systems, we will show that the Lyapunov functions of the energy-second-moment map yields information on the irregularity of the particular phase-space trajectory. Our results will be illustrated by means of numerical examples.

Nonuniform Exponential Dichotomies and Lyapunov Regularity

The notion of exponential dichotomy plays a central role in the Hadamard–Perron theory of invariant manifolds for dynamical systems. The more general notion of nonuniform exponential dichotomy plays a similar role under much weaker assumptions. On the other hand, for nonautonomous linear equations v′ = A(t)v with global solutions, we show here that this more general notion is in fact as weak as possible: namely, any such equation possesses a nonuniform exponential dichotomy. It turns out that the construction of invariant manifolds under the existence of a nonuniform exponential dichotomy requires the nonuniformity to be sufficiently small when compared to the Lyapunov exponents. Thus, it is crucial to estimate the deviation from the uniform exponential behavior. This deviation can be measured by the so-called regularity coefficient, in the context of the classical Lyapunov–Perron regularity theory. We obtain here lower and upper sharp estimates for the regularity coefficient, expressed solely in terms of the matrices A(t).

Stable manifolds for nonautonomous equations without exponential dichotomy

We consider nonautonomous ordinary differential equations v ′ = A ( t ) v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v ′ = A ( t ) v + f ( t , v ) , which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v ′ = A ( t ) v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.

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.