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

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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

Las dos caras del protagonista en Los Persas de Esquilo

Aeschylus found a peculiar way of attaching heroic virtues to the powerful Persian people, in order to fashion a suitable rival to Greece. A powerful enemy was not enough; they had to be righteous and conscious of their limits. In this way the defeat would make the winners truly great. Therefore the poet created a double-faced King. The deceased Darius is presented as a hero and invoked as a god: he was well aware of sophrosyne, he respected divinity and everything he did was right. On the other hand, Xerxes, out of hybris, provoked death and destruction, relying solely on his own abilities. The Persian people were not the ones to blame. It was just the leader’s wrong decision. Fromanother perspective, Aeschylus has spread the defeat throughout the play by means of three catalogues. In the anapaestic parodos the chorus displays the powerful army, naming the captains who went to attack Greece: first catalogue. Further on, the messenger proclaims the news of the defeat, and enumerates (302 ff.) a second catalogue of captains, dead ones this time. It is the counterpart of the parodos’ catalogue. The entrance of Xerxes alone (the only chief who escaped death), is followed by a lyric dialogue with the chorus (chorus leader), where the King is forced to list the names of the captains whose death he provoked: third catalogue. A deadly ring attaches parodos and exodus.Esquilo, al presentar a los persas, ha buscado la manera de mostrarlos a los ojos del público con rasgos heroicos, para convertirlos en rivales dignos de los griegos. No era suficiente que el enemigo fuera poderoso; tenía que ser también justo y consciente de sus límites. Solo así su derrota engrandecerá a los vencedores. Para ello el poeta desdobla en dos al rey de los persas. Darío, ya muerto, se nos muestra dibujado como un héroe e invocado como un dios: practicaba la sophrosyne, respetaba a los dioses y no cometió falta alguna. Jerjes, por el contrario, ha incurrido en hybris, confiando en su capacidad, y ha provocado la catástrofe total de su pueblo. En otro aspecto, Esquilo ha ido desvelando poco a poco la vivencia de la derrota por parte del coro, creando un clímax que magnifica el desastre. Gradúa la catástrofe en tres catálogos. En la Párodo anapéstica, asistimos al despliegue del poderoso ejército: el coro menciona uno tras otro a los capitanes que marcharon contra Grecia (primer catálogo). Más adelante el mensajero que anuncia la derrota, enumera (vv. 302 ss.) un segundo catálogo de capitanes -muertos ahoracomo contrapunto al catálogo inicial. La entrada de Jerjes, único vivo entre sus muertos, va seguida de un diálogo lírico en el que el coro/corifeo obliga al rey a dar cuenta de los nombres de capitanes cuya muerte ha provocado: es el tercer catálogo. Un anillo de muerte enlaza así párodo y éxodo