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 appropr
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Abstract 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 longterm 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.
1 Introduction In this work we study the long-term behaviour of the solutions of non-autonomous ordinary differential equations (ODEs for short) and non-autonomous functional differential equations (FDEs for short). In order to unify our theory, we frequently assume a general expression for the FDEs which contains the ODEs models as a particular case. We investigate the structure and the qualitative behaviour of the omega-limit sets of relatively compact trajectories, as well as some properties for the minimal sets they contain. This information becomes essential to understand the local or global behaviour of the solutions. Some recurrence properties on the temporal variation of the FDEs are assumed, and therefore, their solutions induce a skew-product semiflow with a minimal flow S. Novo () R. Obaya Departamento de Matem´atica Aplicada, E.I. Industriales, Paseo del Cauce 59, Universidad de Valladolid, 47011 Valladolid, Spain e-mail: [email protected]; [email protected] A. Capietto et al., Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lecture Notes in Mathematics 2065, DOI 10.1007/978-3-642-32906-7 4, © Springer-Verlag Berlin Heidelberg 2013
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on the base. As it is common in the analysis of non-autonomous dynamical systems, our phase space is a positively invariant closed subset of ˝ X , where ˝ is a compact metric space with a continuous flow describing the dynamics associated to the time-variation of the equation, and X is the Banach or Fr´echet space representing the state space. In the periodic case ˝ is just a circle, but our formulation includes more general cases as almost-periodic and almost automorphic temporal variations of the vector field, with significative interest in practical applications, in which the structure of ˝ becomes more complicated. Thus we study the trajectories of a complete family of FDEs and we transfer information from one equation to the others using topological and ergodic methods. In general, a collective property is easier to recognize than those verified by a single differential equation; this explains some of the advantages of th
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