Weak topologies for Carath\'eodory differential equations. Continuous dependence, exponential Dichotomy and attractors

We introduce new weak topologies and spaces of Carath\'eodory 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 Carath\'eodory vector field $f$ provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limit set 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.Comment: 34 page

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

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

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

Long-term behavior of nonautonomous neutral compartmental systems

The asymptotic behavior of the trajectories of compartmental systems with a general set of admissible initial data is studied. More precisely, these systems are described by families of monotone nonautonomous neutral functional differential equations with nonautonomous operator. We show that the solutions asymptotically exhibit the same recurrence properties as the transport functions and the coefficients of the neutral operator. Conditions for the cases in which the delays in the neutral and non neutral parts are different, as well as for other cases unaddressed in the previous literature are also obtained

Two dynamical approaches to the notion of exponential separation for random systems of delay differential equations

This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in \JM\ et al.~\cite{MiNoOb1}. Two different approaches to its existence are presented. The state space $X$ will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$ and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains a exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.Comment: arXiv admin note: text overlap with arXiv:1705.0131

Some remarks concerning weakly disconjugate linear Hamiltonian systems

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Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations

Producción CientíficaThis paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.2020-01-012020-01-01Ministerio de Economía, Industria y Competitividad - FEDER (Project MTM2015-66330-P