489 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
Asymptotic behaviour for a class of non-monotone delay differential systems with applications
The paper concerns a class of -dimensional non-autonomous delay
differential equations obtained by adding a non-monotone delayed perturbation
to a linear homogeneous cooperative system of ordinary differential equations.
This family covers a wide set of models used in structured population dynamics.
By exploiting the stability and the monotone character of the linear ODE, we
establish sufficient conditions for both the extinction of all the populations
and the permanence of the system. In the case of DDEs with autonomous
coefficients (but possible time-varying delays), sharp results are obtained,
even in the case of a reducible community matrix. As a sub-product, our results
improve some criteria for autonomous systems published in recent literature. As
an important illustration, the extinction, persistence and permanence of a
non-autonomous Nicholson system with patch structure and multiple
time-dependent delays are analysed.Comment: 26 pages, J Dyn Diff Equat (2017
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
LiâYorke 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
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