59 research outputs found
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 provide
bounded pullback attractors for the processes with vector fields in the
alpha-limit set, the omega-limit set or the whole hull of . 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
ProducciĂłn CientĂficaLinear 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
ProducciĂłn CientĂficaWe 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 CarathĂ©odory 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 CarathĂ©odory 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 CarathĂ©odory 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
ProducciĂłn CientĂficaWe introduce new weak topologies and spaces of CarathĂ©odory 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 SackerâSell spectrum. Second, how particular bounded absorbing sets for the process defined by a CarathĂ©odory 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
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.Comment: 29 pages. arXiv admin note: text overlap with arXiv:2401.1770
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.Comment: 26 page
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 will be a separable ordered Banach space with
, dual space and positive cone normal and
reproducing. In both cases, appropriate cooperativity and irreducibility
conditions are assumed to provide a family of generalized Floquet subspaces. If
in addition 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
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