Víctor Mañosa

Adaptive control of a hysteretic structural system

This paper addresses the problem of controlling a second-order uncertain structural hysteretic system. The hysteretic behavior is represented by a Bouc-Wen model whose parameters lie within intervals. It is shown that the nonlinear response of the model acts as a bounded disturbance. This fact is used to elaborate an adaptive controller, to prove the closed-loop stability and to obtain performance bounds.

Dynamic properties of the hysteretic Bouc-Wen model ☆

Abstract The Bouc-Wen model, widely used in structural and mechanical engineering, gives an analytical description of a smooth hysteretic behavior. In practice, the Bouc-Wen model is mostly used within the following black-box approach: given a set of experimental input–output data, how to adjust the Bouc-Wen model parameters so that the output of the model matches the experimental data. It may happen that a Bouc-Wen model presents a good matching with the experimental real data for a specific input, but does not necessarily keep significant physical properties which are inherent to the real data, independently of the exciting input. This paper presents a characterization of the different classes of Bouc-Wen models in terms of their bounded input-bounded output stability property, and their capability for reproducing physical properties inherent to the true system they are to model.

The focus-centre problem for a type of degenerate system

We consider differential systems in the plane defined by the sum of two homogeneous vector fields. We assume that the origin is a degenerate singular point for these differential systems. We characterize when the singular point is of focus-centre type in a generic case. The problem of its local stability is also considered. We compute the first generalized Lyapunov constant when some non-degeneracy conditions are assumed.

ON THE CENTER PROBLEM FOR DEGENERATE SINGULAR POINTS OF PLANAR VECTOR FIELDS

The center problem for degenerate singular points of planar systems (the degenerate-center problem) is a poorly-understood problem in the qualitative theory of ordinary differential equations. It may be broken down into two problems: the monodromy problem, to decide if the singular point is of focus-center type, and the stability problem, to decide whether it is a focus or a center. We present an outline on the status of the center problem for degenerate singular points, explaining the main techniques and obstructions arising in the study of the problem. We also present some new results. Our new results are the characterization of a family of vector fields having a degenerate monodromic singular point at the origin, and the computation of the generalized first focal value for this family V1. This gives the solution of the stability problem in the monodromic case, except when V1 = 1. Our approach relies on the use of the blow-up technique and the study of the blow-up geometry of singular points. The knowledge of the blow-up geometry is used to generate a bifurcation of a limit cycle.

DYNAMICS OF SOME RATIONAL DISCRETE DYNAMICAL SYSTEMS VIA INVARIANTS

We consider several discrete dynamical systems for which some invariants can be found. Our study includes complex Mobius transformations as well as the third-order Lyness recurrence.

Global periodicity and complete integrability of discrete dynamical systems

Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F p (x)=x for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.

Studying discrete dynamical systems through differential equations

Abstract In this paper we consider dynamical systems generated by a diffeomorphism F defined on U an open subset of R n , and give conditions over F which imply that their dynamics can be understood by studying the flow of an associated differential equation, x ˙ = X ( x ) , also defined on U . In particular the case where F has n − 1 functionally independent first integrals is considered. In this case X is constructed by imposing that it shares with F the same set of first integrals and that the functional equation μ ( F ( x ) ) = det ( D F ( x ) ) μ ( x ) , x ∈ U , has some non-zero solution, μ. Several examples for n = 2 , 3 are presented, most of them coming from several well-known difference equations.

A Darboux-type Theory of Integrability for Discrete Dynamical Systems

There is a method for searching first integrals for polynomial ordinary differential equations (usually called Darboux method) based on the knowledge of several of their invariant algebraic hypersurfaces. We extend this method to discrete dynamical systems, providing a way of searching invariants for them and we give several examples of application.

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

This paper studies non-autonomous Lyness-type recurrences of the form x n+2 = (a n + x n+1)/x n , where {a n } is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k ∈ {1, 2, 3, 6}, the behaviour of the sequence {x n } is simple (integrable), while for the remaining cases satisfying this behaviour can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some different features.

On 2- and 3-periodic Lyness difference equations

We describe the sequences given by the non-autonomous second-order Lyness difference equations , where is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions are also positive. We also show an interesting phenomenon of the discrete dynamical systems associated with some of these difference equations: the existence of one oscillation of their associated rotation number functions. This behaviour does not appear for the autonomous Lyness difference equations.