Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations

Producción CientíficaLinear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as well as on a space of p-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.NCN grant Maestro 2013/08/A/ST1/00275MICIIN/FEDER Grant RTI2018-096523-B-100H2020-MSCA-ITN-2014 643073 CRITICS

Li–Yorke chaos in nonautonomous Hopf bifurcation patterns - I

We analyze the characteristics of the global attractor of a type of dissipative nonautonomous dynamical systems in terms of the Sacker and Sell spectrum of its linear part. The model gives rise to a pattern of nonautonomous Hopf bifurcation which can be understood as a generalization of the classical autonomous one. We pay special attention to the dynamics at the bifurcation point, showing the possibility of occurrence of Li-Yorke chaos in the corresponding attractor and hence of a high degree of unpredictability.MINECO/FEDER, MTM2015-66330-PEuropean Commission, H2020-MSCA-ITN-201

Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics

Producción CientíficaWe study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carathéodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carathéodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carathéodory delay differential equations when the considered phase space is a Sobolev space.MINECO/FEDER MTM2015-66330-PH2020-MSCA-ITN-2014 643073 CRITICS

Non-Atkinson perturbations of nonautonomous linear Hamiltonian systems: exponential dichotomy and nonoscillation

Producción CientíficaWe analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed family satisfies the properties of exponential dichotomy and global existence of $M^+$, then these two properties persist in a neighborhood of 0 which agrees either with the whole real line or with an open negative half-line; and in this last case, the ED fails at the right end value. The properties of ED and of global existence of $M^+$ are fundamental to guarantee the solvability of classical minimization problems given by linear-quadratic control processes.MINECO/FEDER, MTM2015-66330-PEuropean Commission, H2020-MSCA-ITN-201

Weak topologies for Carathéodory differential equations: continuous dependence, exponential dichotomy and attractors

Producción CientíficaWe introduce new weak topologies and spaces of Carathéodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker–Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Carathéodory vector field f provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limitset or the whole hull of f. Conditions for the existence of a pullback or a global attractor for the skew-product semiflow, as well as application examples are also given.MINECO/FEDER Grant MTM2015-66330-PH2020-MSCA-ITN-2014 643073 CRITIC

Characterization of cocycles attractors for nonautonomous reaction-diffusion equations

Producción CientíficaIn this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutioMINECO/FEDER MTM2015-6633

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

Producción CientíficaWe determine sufficient conditions for uniform and strict persistence in the case of skew-product semiflows generated by solutions of non-autonomous families of cooperative systems of ODEs or delay FDEs in terms of the principal spectrums of some associated linear skew-product semiflows which admit a continuous separation. Our conditions are also necessary in the linear case. We apply our results to a noncooperative almost periodic Nicholson system with a patch structure, whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution.MINECO/FEDER MTM2015-6633

Favard Theory and fredholm alternative for disconjugate recurrent second order equations

Producción CientíficaWe discuss the existence of a Fredholm--type Alternative for a recurrent second order linear equation, which is disconjugate in a strong sense. The basic result is about bounded solutions of equations with bounded coefficients: it depends on kinematic similarities that allow to reduce the problem to a pair of very simple normal forms. Then the result is specialized to recurrent equations, by means of Favard theory.MINECO/FEDER MTM2015-6633

Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay

Producción CientíficaThis paper provides a dynamical frame to study non- autonomous parabolic partial differential equations with finite delay. Assuming monotonicity of the linearized semiflow, conditions for the existence of a continuous separation of type II over a minimal set are given. Then, practical criteria for the uniform or strict persistence of the systems above a minimal set are obtained.MINECO/FEDER grant MTM2015-66330-PEuropean Commission under project H2020-MSCA-ITN-2014 643073 CRITIC

Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson systems

Producción CientíficaUsing techniques of non-autonomous dynamical systems, we completely characterize the persistence properties of an almost periodic Nicholson system in terms of some numerically computable exponents. Although similar results hold for a class of cooperative and sublinear models, in the general nonautonomous setting one has to consider persistence as a collective property of the family of systems over the hull: the reason is that uniform persistence is not a robust property in models given by almost periodic differential equations.MINECO / FEDER grant MTM2015-66330-