Bishnani, Zahir (1997) Safety criteria for aperiodic dynamical systems. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Dynamics, Equations, Algebra
Official Date:	July 1997
Dates:	Date Event July 1997 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	MacKay, Robert S., 1956-
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC) ; University of Cambridge ; British Council
Extent:	v, 129 leaves : illustrations
Language:	eng

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