10,798 research outputs found
Dichotomies for evolution equations in Banach spaces
The aim of this paper is to emphasize various concepts of dichotomies for
evolution equations in Banach spaces, due to the important role they play in
the approach of stable, instable and central manifolds. The asymptotic
properties of the solutions of the evolution equations are studied by means of
the asymptotic behaviors for skew-evolution semiflows.Comment: 22 page
Exponential dichotomies of evolution operators in Banach spaces
This paper considers three dichotomy concepts (exponential dichotomy, uniform
exponential dichotomy and strong exponential dichotomy) in the general context
of non-invertible evolution operators in Banach spaces. Connections between
these concepts are illustrated. Using the notion of Green function, we give
necessary conditions and sufficient ones for strong exponential dichotomy. Some
illustrative examples are presented to prove that the converse of some
implication type theorems are not valid
Lyapunov theorems for Banach spaces
We present a spectral mapping theorem for semigroups on any Banach space .
From this, we obtain a characterization of exponential dichotomy for
nonautonomous differential equations for -valued functions. This
characterization is given in terms of the spectrum of the generator of the
semigroup of evolutionary operators.Comment: 6 page
On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems
For nonlinear time-delay systems, domains of attraction are rarely studied
despite their importance for technological applications. The present paper
provides methodological hints for the determination of an upper bound on the
radius of attraction by numerical means. Thereby, the respective Banach space
for initial functions has to be selected and primary initial functions have to
be chosen. The latter are used in time-forward simulations to determine a first
upper bound on the radius of attraction. Thereafter, this upper bound is
refined by secondary initial functions, which result a posteriori from the
preceding simulations. Additionally, a bifurcation analysis should be
undertaken. This analysis results in a possible improvement of the previous
estimation. An example of a time-delayed swing equation demonstrates the
various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in
'Nonlinear Dynamics'. The final authenticated version is available online at
https://doi.org/10.1007/s11071-020-05620-8
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