Rafael Obaya

Exponential dichotomy and trichotomy for difference equations

Abstract In this paper, a roughness theorem of exponential dichotomy and trichotomy of linear difference equations is proved. It is also shown that if an almost periodic difference equation has an exponential dichotomy on a sufficiently long finite interval, then it has one on (−∞, +∞).

OLD AND NEW RESULTS ON STRANGE NONCHAOTIC ATTRACTORS

Classical and new results concerning the topological structure of skew-products semiflows, coming from nonautonomous maps and differential equations, are combined in order to establish rigorous conditions giving rise to the occurrence of strange nonchaotic attractors on 𝕋d × ℝ. A special attention is paid to the relation of these sets with the almost automorphic extensions of the base flow. The scope of the results is clarified by applying them to the Harper map, although they are valid in a much wider context.

Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences

Abstract The existence of almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions of differential equations with piecewise constant argument is characterized in terms of almost periodic, asymptotically, and pseudo almost periodic sequences. Thus Meisterss and Opials theorems are extended.

Strictly ordered minimal subsets of a class of convex monotone skew-product semiflows

We study the topological and ergodic structure of a class of convex and monotone skew-product semiflows. We assume the existence of two strongly ordered minimal subsets K1,K2 and we obtain an ergodic representation of their upper Lyapunov exponents. In the case of null upper Lyapunov exponents, we obtain a lamination into minimal subsets of an intermediate region where the restriction of the semiflow is affine. In the hyperbolic case, we deduce the long-time behaviour of every trajectory ordered with K2. Some examples of skew-product semiflows generated by non-autonomous differential equations and satisfying the assumptions of monotonicity and convexity are also presented.

An Ergodic Classification of Bidimensional Linear Systems

The ergodic structure of the projective flow induced by a family of bidimensional linear systems is studied. It is shown that the existence of a continuous invariant measure guarantees the existence of another measure, called linear by the authors, which provides substantial information upon the properties of the complex bundle. Some examples are given to illustrate the applicability of these results.

Minimal sets in monotone and sublinear skew-product semiflows II: Two-dimensional systems of differential equations

Article history: Received 18 June 2009 Revised 2 December 2009 Available online 8 February 2010 MSC: 37B55 37C65 34C12 39B99 35K57

Uniform Weak Disconjugacy and Principal Solutions for Linear Hamiltonian Systems

The paper analyzes the property of (uniform) weak disconjugacy for nonautonomous linear Hamiltonian systems, showing that it is a convenient replacement for the more restrictive property of disconjugacy. In particular, its occurrence ensures the existence of principal solutions. The analysis of the properties of these solutions provides ample information about the dynamics induced by the Hamiltonian system on the Lagrange bundle.

EXPONENTIAL TRICHOTOMY AND A CLASS OF ERGODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH ERGODIC PERTURBATIONS

Abstract The existence of a class of ergodic solutions of some differential equations is investigated by using exponential trichotomy. An application to the Hill equation with ergodic forcing function is given.

Nonautonomous linear Hamiltonian systems oscillation, spectral theory and control

Nonautonomous linear Hamiltonian systems.- The rotation number and the Lyapunov index for real nonautonomous linear Hamiltonian systems.- The Floquet coeffcient for nonautonomous linear Hamiltonian systems: Atkinson problems.- The Weyl functions.- Weak disconjugacy for linear Hamiltonian systems.- Nonautonomous control theory. Linear regulator problem and the Kalman-Bucy filter.- Nonautonomous control theory. A general version of the Yakubovich Frequency Theorem.- Nonautonomous control theory. Linear-quadratic dissipative control processes.- Index.- References

Attractor minimal sets for non-autonomous delay functional differential equations with applications for neural networks

The dynamics of a class of non-autonomous, convex (or concave) and monotone delay functional differential systems is studied. In particular, we provide an attractivity result when two completely strongly ordered minimal subsets K1≪CK2 exist. As an application of our results, sufficient conditions for the existence of global or partial attractors for non-autonomous delayed Hopfield-type neural networks are obtained.