133 research outputs found
Admissibility and general dichotomies for evolution families
For an arbitrary noninvertible evolution family on the half-line and for
in a large class of rate functions, we
consider the notion of a -dichotomy with respect to a family of norms and
characterize it in terms of two admissibility conditions. In particular, our
results are applicable to exponential as well as polynomial dichotomies with
respect to a family of norms. As a nontrivial application of our work, we
establish the robustness of general nonuniform dichotomies
Robustness of nonuniform and random exponential dichotomies with applications to differential equations
In this thesis, we study hyperbolicity for deterministic and random
nonautonomous dynamical systems and their applications to differential
equations. More precisely, we present results in the following topics:
nonuniform hyperbolicity for evolution processes and hyperbolicity for
nonautonomous random dynamical systems. In the first topic, we study
the robustness of the nonuniform exponential dichotomy for continuous
and discrete evolution processes. We present an example of an infinitedimensional
differential equation that admits a nonuniform exponential
dichotomy and apply the robustness result. Moreover, we study the
persistence of nonuniform hyperbolic solutions in semilinear differential
equations. Furthermore, we introduce a new concept of nonuniform exponential
dichotomy, provide examples, and prove a stability result under
perturbations for it. In the second topic, we introduce exponential dichotomies
for random and nonautonomous dynamical systems. We prove
a robustness result for this notion of hyperbolicity and study its applications
to random and nonautonomous differential equations. Among these
applications, we study the existence and continuity of random hyperbolic
solutions and their associated unstable manifolds. As a consequence, we
obtain continuity and topological structural stability for nonautonomous
random attractors
Safety criteria for aperiodic dynamical systems
The
use of
dynamical
system models
is
commonplace
in
many areas of science and
engineering.
One is
often
interested in
whether
the
attracting solutions
in these
models are
robust
to perturbations of
the
equations of motion.
This
question
is
extremely
important
in
situations where
it is
undesirable
to have
a
large
response
to
perturbations
for
reasons
of safety.
An
especially
interesting
case occurs when the
perturbations are aperiodic and
their
exact
form is
unknown.
Unfortunately,
there is
a
lack
of
theory in the literature that
deals
with
this
situation.
It
would
be
extremely useful to have
a practical
technique that
provides
an upper
bound
on the size of the
response
for
an arbitrary perturbation of given
size.
Estimates
of
this form
would allow the
simple
determination
of safety criteria
that
guarantee
the response
falls
within some pre-specified safety
limits. An
excellent area
of application
for this technique
would
be
engineering systems.
Here
one
is frequently
faced
with
the
problem of obtaining safety criteria
for
systems
that in
operational use are
subject
to unknown, aperiodic perturbations.
In this thesis I
show
that
such safety criteria are easy to obtain
by
using
the
concept
of persistence
of
hyperbolicity. This
persistence result
is
well
known in the theory
of
dynamical systems.
The formulation I
give
is functional
analytic
in
nature and
this has
the
advantage
that it is
easy
to
generalise and
is
especially suited to the
problem of
unknown,
aperiodic perturbations.
The
proof
I
give of
the
persistence
theorem
provides
a
technique
for
obtaining
the
safety estimates we want and
the
main part of
this thesis is
an
investigation into how this
can
be
practically
done.
The
usefulness of
the technique is illustrated through two
example systems,
both
of
which are
forced
oscillators.
Firstly, I
consider
the
case where
the
unforced oscillator
has
an asymptotically stable equilibrium.
A
good application of this is the
problem of
ship stability.
The
model
is
called
the
escape equation and
has been
argued to
capture
the relevant
dynamics
of a ship at sea.
The
problem is to find
practical criteria
that
guarantee
the
ship
does not capsize or go
through large
motions when there are external
influences like
wind and waves.
I
show
how
to
provide good criteria which ensure a safe
response when
the
external
forcing is
an arbitrary,
bounded function
of
time. I
also
consider
in
some
detail the
phased-locked loop. This is
a periodically forced
oscillator
which
has
an attracting periodic solution that is
synchronised
(or
phase-locked) with
the
external
forcing. It is interesting to
consider the
effect of small aperiodic variations
in the
external
forcing. For hyperbolic
solutions
I
show that the
phase-locking persists and
I
give
a method
by
which one can
find
an upperbound
on
the
maximum size of
the
response
Generalized Trichotomies: robustness and global and local invariant manifolds
In a Banach space, given a differential equation v′(t) = A(t)v(t), with an initial
condition v(s) = vs and that admits a generalized trichotomy, we studied
which type of conditions we need to impose to the linear perturbations B so that
v′(t) = [A(t) + B(t)] v(t) continues to admit a generalized trichotomy, that is, we
studied the robustness of generalized trichotomies. In the same way, it was also the
aim of our work the study of a differential equation with another type of nonlinear
perturbations, v′(t) = A(t)v(t) + f(t, v). We sought conditions to impose on
the function f so that the new perturbed equation would admit a global Lipschitz
invariant manifold as well as the necessary conditions for the existence of local Lipschitz
invariant manifolds.Num espaço de Banach, dada uma equação diferencial v′(t) = A(t)v(t), sujeita
a uma condição inicial v(s) = vs e que admite uma tricotomia generalizada, estudámos o tipo de condições a impor às perturbações lineares B de modo que a equação v′(t) = [A(t) + B(t)] v(t) ainda admita uma tricotomia generalizada, ou seja, estudámos a robustez das tricotomias generalizadas. Da mesma forma, foi também objecto deste trabalho, o estudo de uma equação diferencial com outro tipo de perturbações não lineares, v′(t) = A(t)v(t) + f(t, v). Procurámos condições necessárias a impor à função f por forma a que a nova equação perturbada admitisse uma variedade invariante Lipschitz global, bem como as condições necessárias para a existência de variedades invariantes Lipschitz locais
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