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In this paper we introduce a new method for obtaining boundedness of solutions of integral equations. From the integral equation we form an integrodifferential equation by computing x' + kx to which we apply a Liapunov functional. This can be far more effective than the usual technique of differentiating the equation. The qualitative properties derived from that equation are then transferred to a majorizing function for the integral equation. Schaefer's fixed point theorem is used to conclude that there is a periodic solution. Three kinds of integral equations are studied and they are shown to be related through two examples.En este artículo presentamos un nuevo método para obtener acotación de soluciones de ecuaciones integrales. A partir de la ecuación integral, formamos una ecuación integrales diferencial calculando x' + kx mediante la aplicación de un funcional de Liapunov. Ello puede resultar bastante más efectivo que la técnica usual de diferenciación de la ecuación. Las propiedades cualitativas derivadas de la ecuación son entonces transferidas a la función mayorante para la ecuación integral. El teorema del punto fijo de Schaefer es usado para concluir que hay una solución periódica. Se estudia tres tipos de ecuaciones integrales y se muestra que ellas están relacionadas a través de dos ejemplos.

In this article, we find stability criteria for perturbed differential systems, in terms of two measures. Our main tool is a definition of total stability based on two classes of perturbations.

This paper concerns the analysis of transferring stability properties from an invariant manifold to the whole space for an ordinary differential system. In previous papers we already treated this problem in the case of asymptotic and total stability. Here we deal with the case of non-asymptotic stability. We generalize to differential systems depending on time a reduction principle (Kelley in J Math Anal Appl 18:336-344, ; Pliss in Izv Akad Nauk SSSR Mat Ser 28:1297-1324, ) relative to autonomous systems. Our procedure is very different from the fixed point theorem argument used in Kelley (J Math Anal Appl 18:336-344, ), and it is based on the use of a suitable Liapunov function. Some results concerning integral stability are also given. [ABSTRACT FROM AUTHOR]

This paper uses tools from dynamical systems theory to investigate the properties of Canadian and US money and velocity measures. In doing so, we follow the recent contribution by Whang and Linton [Whang, Y.-J., Linton, O., 1999. The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series. Journal of Econometrics 91, 1-42] and construct the standard error for the Nychka et al. [Nychka, D.W., Ellner, S., Gallant, R.A., McCaffrey, D., 1992. Finding chaos in noisy systems. Journal of the Royal Statistical Society B 54, 399-426] dominant Lyapunov exponent. Comparisons are made among simple-sum, Divisia, and currency equivalent aggregates at different levels of monetary aggregation. We find statistically significant evidence against low-dimensional chaos and point to the use of stochastic models and statistical inference in the modeling of these variables. JEL classification: E40; E50; C32 Keywords: Divisia; Chaos; Lyapunov exponent

Dedicated to Steve Smale on His 75th Birthday Abstract. In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. [...]

In order to be able to successfully apply direct methods or other energy-function-based calculations in power systems, which include flexible AC transmission systems (FACTS) devices, the influence of those devices should be properly considered. This is currently not always possible in power systems incorporating unified power flow controllers (UPFCs) because the present energy functions for electric power systems do not involve proper UPFC actions. This paper presents a way of incorporating the transient-stability augmentation action of the most versatile FACTS device, i.e., the UPFC, into an energy function for multimachine systems. After making some assumptions, a new term of the structure-preserving energy function (SPEF) that represents a UPFC's energy function was constructed. This term can simply be added to any existing SPEF. The extended SPEF was tested for one UPFC in a longitudinal test system and for one and two UPFCs in an IEEE nine-bus machine test system. A comparison between the critical clearing times (CCTs) acquired directly with the use of the newly constructed SPEF and those obtained with time-simulation results shows that the proposed UPFC's energy function is suitable because the correct CCTs were obtained. Index Terms--FACTS devices, Lyapunov methods, power system control, power system transient stability.

We present Lyapunov stability results, including converse theorems, for a class of discontinuous dynamical systems (DDS) determined by the solutions of retarded functional differential equations. We demonstrate the applicability of these results in the analysis of several specific important classes of DDS determined by functional differential equations and differential-difference equations. Index Terms--Asymptotic stability, differential-difference equations, discontinuous dynamical systems (DDS), exponential stability, hybrid systems, Lyapunov stability, retarded functional differential equations, switched systems, systems with delays.

In this paper, we present a delayed neural network approach to solve linear projection equations. The Lyapunov-Krasovskii theory for functional differential equations and the linear matrix inequality (LMI) approach are employed to analyze the global asymptotic stability and global exponential stability of the delayed neural network. Compared with the existing linear projection neural network, theoretical results and illustrative examples show that the delayed neural network can effectively solve a class of linear projection equations and some quadratic programming problems. Index Terms--Asymptotical stability, delayed neural networks, exponential stability, linear matrix inequality (LMI), Lyapunov-Krasovskii functional, quadratic programming.

Let X C [R.sup.N] and consider a system x = f (x, u), f : X X [R.sup.M] [right arrow] [R.sup.N], with the property that the associated autonomous system x = f (x, 0) has an asymptotically stable compactum C with region of attraction A. Assume that x is a solution of the former, defined on [0, [infinity]), corresponding to an input function u. Assume further that, for each compact K C X, there exists k > 0 such that | f (z, v)--f (z, 0) | [less than or equal to] k | v | for all (z, v) [member of] K X [R.sup.M]. A simple proof is given of the following [L.sup.p]-input converging-state property: if u [member of] [L.sup.p] for some p [member of] [1, [infinity]) and x has an [omega]-limit point in A, then x approaches C. Index Terms--Asymptotic stability, converse Lyapunov theory, domain of attraction.

We present some new global stability results of neural networks with delay and show that these results generalize recently published stability results. In particular, several different stability conditions in the literature which were proved using different Lyapunov functionals are generalized and unified by proving them using the same Lyapunov functional. We also show that under certain conditions, reversing the directions of the coupling between neurons preserves the global asymptotical stability of the neural network. Index Terms--Asymptotical stability, delay equations, Lyapunov functional, neural networks.

Delay-dependent/delay-independent uniform asymptotic stability and uniform stability criteria for linear systems with multiple time-varying delays are established respectively in this note. The results are derived based on a new-type stability theorem for retarded dynamical systems and a new analysis technique for estimating the derivative of a Lyapunov function along the solution of a system at certain specific instants. Four remarks together with an illustrative example are given to compare the obtained results with and to show their superiority to those in the literature. Index Terms--Linear systems, Lyapunov methods, stability, time delay.

This note treats the problem of stabilization of linear systems by static output feedback using the concept of (C, A, B)-invariant subspaces. The work provides a new characterization of output stabilizable (C, A, B)-invariant subspaces through two coupled quadratic stabilization conditions. An equivalence is shown between the existence of a solution to this set of conditions and the possibility to stabilize the system by static output feedback. An algorithm is provided and numerical examples are reported to illustrate the approach. Index Terms--Convex programming, geometric approach, Lyapunov equations, output feedback, quadratic stabilizability.

A sufficient condition for robust D-stability of linear systems with polytope type uncertainties is proposed. The result is based on a linear parameter-dependent Lyapunov function obtained from the feasibility test of a set of linear matrix inequalities (LMIs) defined at the vertices of the polytope. This improved LMI condition encompasses previous results based on additional inequalities, as well as results based on extra variables. Index Terms--Linear matrix inequalities (LMIs), parameter-dependent Lyapunov functions, parametric uncertainty, polytopic uncertainty, robust stability.

A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant. Index Terms--Constrained control, invariant set, quadratic functions.

This brief presents three sufficient conditions for the global exponential stability of cellular neural networks with time delays. The new stability results provide algebraic criteria for stability verifications and improve upon exisfing ones with stronger conditions. To demonstrate the differences and features of the new stability criteria, several examples are discussed to compare the present results with the existing ones. Index Terms--Cellular neural networks (CNNs), global exponential stability, Lyapunov function, time delays.

There are, in general, two classes of results regarding the synchronization of chaos in an array of coupled identical chaotic systems. The first class of results relies on Lyapunov's direct method and gives analytical criteria for global or local synchronization. The second class of results relies on linearization around the synchronization manifold and the computation of Lyapunov exponents. The computation of Lyapunov exponents is mainly done via numerical experiments and can only show local synchronization in the neighborhood of the synchronization manifold. On the other hand, Lyapunov's direct method is more rigorous and can give global results. The coupling topology is generally expressed in matrix form and the first class of methods mainly deals with symmetric matrices whereas the second class of methods can work with all diagonalizable matrices. The purpose of this brief is to bridge the gap in the applicability of the two classes of methods by considering the nonsymmetric case for the first class of methods. We derive a synchronization criterion for nonreciprocal coupling related to a numerical quantity that depends on the coupling topology and we present methods for computing this quantity. Index Terms--Chaos, convex programming, coupled oscillators, Lyapunov exponents, Lyapunov functions, nonlinear programming, nonreciprocal coupling, synchronization.

In this article, we find stability criteria for perturbed differential systems, in terms of two measures. Our main tool is a definition of total stability based on two classes of perturbations. [ABSTRACT FROM AUTHOR]

The concept of isles of Eden when generalized from finite cellular automata to classical dynamical systems on ℝd calls the attention to the ZUK Theorem — Z for [Zubov, 1957], U and K for [Ura & Kimura, 1960] — a fifty years old and almost forgotten result in topological dynamics, a classification theorem for compact isolated invariant sets. We provide a simple proof and discuss various consequences including a new result on pointwise Liapunov functions. [ABSTRACT FROM AUTHOR]

Abstract: Some sufficient conditions are derived to ensure the global exponential stability of delayed bi-directional associative memory (BAM) neural network on time scale, using the time scale calculus theory and the Liapunov functional method. The conditions possess highly important significance and can be easily checked in practice by simple algebraic methods. This is the first time applying the time scale calculus theory to unify and improve discrete-time and continuous-time bi-directional associate memory neural network under the same framework. [Copyright &y& Elsevier]

The concept of isles of Eden when generalized from finite cellular automata to classical dynamical systems on ℝd calls the attention to the ZUK Theorem — Z for [Zubov, 1957], U and K for [Ura & Kimura, 1960] — a fifty years old and almost forgotten result in topological dynamics, a classification theorem for compact isolated invariant sets. We provide a simple proof and discuss various consequences including a new result on pointwise Liapunov functions. [ABSTRACT FROM AUTHOR]

We create real valued continuous functions which decrease on orbits of relations. This will generalize the notion of Liapunov functions to the more abstract setting of closed relations on compact Hausdorff spaces. [ABSTRACT FROM AUTHOR]

Abstract: The defining property of an integral equation with resolvent is the relation between and for functions in a given vector space. We study the behaviour of a solution of an integral equation: when is periodic, , while is typified by with . There is a resolvent, , so that We show that the integral so closely approximates that the only trace of that large function, , in the solution is an -function, . In short, that large function has essentially no long-term effect on the solution which turns out to be the sum of a periodic function, a function tending to zero, and an -function. The noteworthy property here is that with great precision the integral can duplicate vector spaces of functions both large and small, both monotone and oscillatory; however, it cannot duplicate a given nontrivial periodic function other than where is constant. The integral is an approximation to for , but contraction mappings show us that precisely at that approximation fails and approaches a nontrivial periodic function. [Copyright &y& Elsevier]

In this paper we study the existence, uniqueness and stability of periodic solutions for a two-neuron network system with or without external inputs. The system consists of two identical neurons, each possessing nonlinear feedback and connected to the other neuron via a nonlinear sigmoidal activation function. In the absence of external inputs but with appropriate conditions on the feedback and connection strengths, we prove the existence, uniqueness and stability of periodic solutions by using the Poincaré–Bendixson theorem together with Dulac's criterion. On the other hand, for the system with periodic external inputs, combining the techniques of the Liapunov function with the contraction mapping theorem, we propose some sufficient conditions for establishing the existence, uniqueness and exponential stability of the periodic solutions. Some numerical results are also provided to demonstrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this work, a new approach to stability theory of impulsive functional differential equations is proposed. Instead of putting all components of the state variable x in one Liapunov function, several functions of partial components of x, which can be much easier constructed, are used so that the conditions ensuring that stability are simpler and less restrictive. Also, an example is given to illustrate the advantages of the obtained results. [Copyright &y& Elsevier]

This paper analyzes local and global behavior of several dynamical systems which generalize some artificial neural network (ANN) semilinear models originally designed for principal component analysis (PCA) in the characterization of random vectors. These systems implicitly performed the spectral analysis of correlation (i.e. symmetric positive definite) matrices. Here, the proposed generalizations cover both nonsymmetric matrices as well as fully nonlinear models. Local stability analysis is performed via linearization and global behavior is analyzed by constructing several Liapunov functions. [Copyright &y& Elsevier]

We are concerned with plane differential systems of the formx˙ = P(x,y), y˙ = Q(x,y), withP, Q analytic. We propose a formal-numeric method to localize the attractors and the repellers of the system. Such a method consists of looking for a power series solution to a PDE of the typePSHAPE="BUILT" ALIGN="C" STYLE="S">∂V∂x + QSHAPE="BUILT" ALIGN="C" STYLE="S">∂V∂y = μ(V), with μ is an arbitrary analytic function. Whenμ(V) = ρ(V − V2),ρ > 0, the attractors are contained in the setV = 1, the repellers in the setV = 0. [Copyright &y& Elsevier]

The main result of this paper is a converse Liapunov theorem which applies to systems of ordinary differential equations with a discontinuous righthand side. We treat both the problem of local stability of an equilibrium position and the problem of boundedness of solutions. In particular, we show that in order to achieve a necessary and sufficient condition in terms of continuous Liapunov functions, the classical definitions need to be strengthened in a convenient way. This work was motivated by the recently renewed interest in stabilization by discontinuous feedback and analysis of the state evolution with respect to bounded inputs. To achieve a more general treatment, the exposition is developed in the framework of differential inclusions theory. [ABSTRACT FROM AUTHOR]

A robust regulator for flexible-joint robots is proposed, which yields constant torque disturbance rejection acting on the links. The design uses the integrator backstepping technique [4,5] to cancel nonlinearities and disturbance not in the range space of the control. Stability of the closed loop system is shown using iterative Liapunov functions. [ABSTRACT FROM AUTHOR]

We present some new results which definitively explain thebehavior of the classical, heuristic nonlinear relaxation labelingalgorithm of Rosenfeld, Hummel, and Zucker in terms of theHummel-Zucker consistency theory and dynamical systems theory. Inparticular, it is shown that, when a certain symmetry condition is met,the algorithm possesses a Liapunov function which turns out to be (thenegative of) a well-known consistency measure. This follows almostimmediately from a powerful result of Baum and Eagon developed in thecontext of Markov chain theory. Moreover, it is seen that most of theessential dynamical properties of the algorithm are retained when thesymmetry restriction is relaxed. These properties are also shown tonaturally generalize to higher-order relaxation schemes. Someapplications and implications of the presented results are finallyoutlined. [ABSTRACT FROM AUTHOR]

This comment on the Serletis and Shintani paper in this journal issue emphasizes misunderstandings and oversimplifications that exist within the literature, regarding the relevancy and importance of chaos in economics. The paper by Serletis and Shintani provides substantial insights into that frequently misunderstood literature. JEL classification: E40; E50; C32 Keywords: Divisia; Chaos; Lyapunov exponent

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered is X′=f(t,X) and the associated perturbed differential equation is Y′=f(t,Y)+g(t,Y).

A new result for bounding the summations for solution eigen-values of the algebraic Lyapunov equation is presented. This makes possible to generate the best known upper scalar solution estimates. Index Terms--Lyapunov equation, matrix measure, solution bounds, trace.