Sylvia Novo

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.

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.

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.

Almost periodic and almost automorphic dynamics for scalar convex differential equations

We study the set of bounded trajectories for the flow defined by a class of scalar convex differential equations depending on a parameter. It is found that there exists precisely one value of the parameter for which almost automorphic but not almost periodic dynamics may appear. Even for this parameter value, the occurrence of almost periodic dynamics is shown to be residual in some cases. The dependence of this parameter on the functions defining the differential equations is also studied.

Neutral Functional Differential Equations with Applications to Compartmental Systems

We study the monotone skew-product semiflow generated by a family of neutral functional differential equations with infinite delay and stable D-operator. The stability properties of D allow us to introduce a new order and to take the neutral family to a family of functional differential equations with infinite delay. Next, we establish the 1-covering property of omega-limit sets under the componentwise separating property and uniform stability. Finally, the obtained results are applied to the study of the long-term behavior of the amount of material within the compartments of a neutral compartmental system with infinite delay.

Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows

Several results of uniform persistence above and below a minimal set of an abstract monotone skew-product semiflow are obtained. When the minimal set has a continuous separation the results are given in terms of the principal spectrum. In the case that the semiflow is generated by the solutions of a family of non-autonomous differential equations of ordinary, delay or parabolic type, the former results are strongly improved. A method of calculus of the upper Lyapunov exponent of the minimal set is also determined.

Non-autonomous Functional Differential Equations and Applications

This chapter deals with the applications of dynamical systems techniques to the study of non-autonomous, monotone and recurrent functional differential equations. After introducing the basic concepts in the theory of skew-product semiflows and the appropriate topological dynamics techniques, we study the long-term behavior of relatively compact trajectories by describing the structure of minimal and omega-limit sets, as well as the attractors. Both the cases of finite and infinite delay are considered. In particular, we show the relevance of uniform stability in this study. Special attention is also paid to the almost periodic case, in which the presence of almost periodic and almost automorphic dynamics is analyzed. Some applications of these techniques to the study of neural networks, compartmental systems and certain biochemical control circuit models are shown.

Exponential Ordering for Nonautonomous Neutral Functional Differential Equations

We study monotone skew-product semiflows generated by families of nonautonomous neutral functional differential equations with infinite delay and stable D-operator, when the exponential ordering is considered. Under adequate hypotheses of stability for the order on bounded sets, we show that the omega-limit sets are copies of the base to explain the long-term behavior of the trajectories. The application to the study of the amount of material within the compartments of a neutral compartmental system with infinite delay shows the improvement with respect to the standard ordering.