1,435 research outputs found
Existence of global attractor for a nonautonomous state-dependent delay differential equation of neuronal type
The analysis of the long-term behavior of the mathematical model of a neural
network constitutes a suitable framework to develop new tools for the dynamical
description of nonautonomous state-dependent delay equations (SDDEs).
The concept of global
attractor is given, and some results which establish properties ensuring
its existence and providing a description of its shape, are proved.
Conditions for the exponential stability of the global attractor
are also studied. Some properties
of comparison of solutions constitute a key in
the proof of the main results, introducing methods of monotonicity
in the dynamical analysis of nonautonomous SDDEs.
Numerical simulations of some illustrative models show
the applicability of the theory.Ministerio de EconomĂa y Competitividad / FEDER, MTM2015-66330-PMinisterio de Ciencia, InnovaciĂłn y Universidades, RTI2018-096523-B-I00European Commission, H2020-MSCA-ITN-201
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
The random case of Conley's theorem: III. Random semiflow case and Morse decomposition
In the first part of this paper, we generalize the results of the author
\cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we
obtain Conley decomposition theorem for infinite dimensional random dynamical
systems. In the second part, by introducing the backward orbit for random
semiflow, we are able to decompose invariant random compact set (e.g. global
random attractor) into random Morse sets and connecting orbits between them,
which generalizes the Morse decomposition of invariant sets originated from
Conley \cite{Con} to the random semiflow setting and gives the positive answer
to an open problem put forward by Caraballo and Langa \cite{CL}.Comment: 21 pages, no figur
Recent developments in dynamical systems: three perspectives
This paper aims to an present account of some problems considered in the past years in Dynamical Systems, new research directions and also provide some open problems
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