Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
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Lectures
on
Dynamical
Systems,
Structural
Stability
and
their
Applications
KOTIKK.
LEE
Dept.
of
Electrical
and
Computer Engineering
University
of
Colorado
'
h
World Scientific
f"
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P 0 Box 128, Farrer Road, Singapore 9128
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office:
73
Lynton Mead, Toueridge, London N20 8DH
LECTURES
ON DYNAMICAL SYSTEMS,
STRUCTURAL
STABILITY AND
THEIR
APPLICATIONS
Copyright© 1992
by
World Scientific Publishing Co. Pte. Ltd.
All
rights reserved. This book, or parts thereof. may not be reproduced
in
any form
or
by any means, electronic or mechanical, including photocopying, recording
or
any
information storage and retrieval system now known or to be invented, without
wrillen permission from the Publisher.
ISBN 9971-50-965-2
Printed
in
Singapore
by
JBW Printers and Binders Pte. Ltd.
Dedicated
to
Peter
G.
Bergmann
and
Heinz
Helfenstein,
who
taught
me
physics
and
mathematics,
and
to
Lydia,
together
they
taught
me
humanity.
v
PREFACE
In
this
past
decade,
we
have witnessed
the
enormous
growth
of
an
interdisciplinary
field
of
study,
namely,
dynamical systems. Yet, dynamical systems,
as
a
subfield
of
mathematics, has been
established
since
late
last
century
by
Poincare [1881], and
reinforced
by
Liapunov (1892].
Parts
of
such a surge
are
due
to
many
major advances
in
differential
topology,
the
geometric
theory
of
differential
equations,
algebraic
geometry,
nonlinear
functional
analysis,
and
nonlinear
global
analysis,
just
to
name
a few. This
is
partly
due
to
computers being
readily
available.
Nowadays,
any second-year
college
student
can use a
personal
computer
to
program
the
evolution
of
a
nonlinear
difference
equation
and
finq
all
kinds
of
chaotic
behavior. With
the
new
level
of
sophistication
of
graphics,
it
can evolve
into
"mathematical
entertainment"
or
"arts".
Each popular
science
or
engineering magazine has
at
least
one
article
on
nonlinear
dynamical systems a
year.
It
not
only caught
the
fascination
of
the
scientists,
but
also
attracted
the
attention
of
the
enlightened
public.
Ten
to
fifteen
years
ago,
there
were
but
a dozen papers
on
dynamical systems
in
each major mathematics
or
physics
journal
per
year.
At
present,
the
majority
of
these
journals
contain
sections
on
nonlinear
sciences,
and indeed
there
are
several
new
journals
solely
devoted
to
this
subject.
Nonetheless,
the
communication and
infusion
of
knowledge between
the
mathematicians working
on
the
analytic
approach and
the
scientists
and
engineers
working mostly
on
the
applications
and numerical
simulations
have been
less
than
ideal.
Part
of
the
reason
is
cultural.
Mathematicians
tend
to
approach
the
problem
in
a
more
generic
sense,
that
is,
they
tend
to
ask
questions
and look
for
answers
for
more
general
properties
and
the
underlying
structures
of
the
systems.
Books
written
by mathematicians
usually
treat
the
subject
vii
with
mathematical
rigor,
but
lack
of
some
motivation
for
wanting
to
study
the
underlying
mathematical
structures
of
the
system.
The
treatment
by
scientists
and
engineers
usually
encompasses
too
many
details
and
misses
the
underlying
structures
which
may
transcend
the
usual
boundaries
of
various
disciplines.
Thus,
scientists
and
engineers
may
not
be
aware
of
the
advances
of
the
same
type
of
problems
in
other
disciplines.
We
shall
give
examples
of
such
underlying
mathematical
structures
for
diverse
disciplines.
This
volume
intends
to
bridge
the
gap
between
these
two
categories
of
books
treating
nonlinear
dynamical
systems.
Thus,
we
would
like
to
bridge
the
gap
and
foster
communication
between
scientists
and
mathematicians.
In
the
following,
proofs
of
theorems
are
usually
not
given.
Instead,
examples
are
provided
so
that
the
readers
can
get
the
meaning
of
the
theorems
and
definitions.
In
other
words,
we
would
like
the
readers
to
get
some
sense
of
the
concepts
and
techniques
of
the
mathematics
as
well
as
its
"culture".
This
volume
is
based
on
the
lecture
notes
of
a
graduate
course
I
gave
in
1983-4
while
I
was
with
TRW.
Chapter
1
introduces
the
concept
of
dynamical
systems
and
stability
with
examples
from
physics,
biology,
and
economics.
We
want
to
point
out
that
even
though
the
differential
or
difference
equations
governing
those
phenomena
are
different,
nonetheless,
the
mathematical
procedures
in
analyzing
them
are
the
same.
We
also
try
to
motivate
the
reader
about
the
concept
and
the
need
to
study
the
structural
stability.
In
Chapter
2,
we
assemble
most
of
the
definitions
and
theorems
about
basic
properties
of
algebra,
points
set
(also
called
general)
topology,
algebraic
and
differential
topology,
differentiable
manifolds
and
differential
geometry,
which
are
needed
in
the
course
of
the
lectures.
Our
main
purpose
is
to
establish
notations,
terminology,
concepts
and
structures.
We
provide
definitions,
examples,
and
essential
results
without
giving
the
arguments
of
proof.
The
material
presented
is
slightly
more
than
absolutely
viii
necessary
for
the
course
of
this
lecture
series.
For
instance,
we
certainly
can
get
by
without
explicitly
talking
about
algebraic
topology,
such
as
homotopy
and
homology.
Nonetheless,
we
do
utilize
the
concepts
of
orientable
manifolds
or
spaces
so
that
an
everywhere
non-zero
volume
element
can
be
found.
We
also
discuss
the
connected
components
and
connected
sums
of
a
state
space.
Nor
do
we
have
to
discuss
tubular
neighborhoods,
even
though
when
dealing
with
return
maps,
the
stability
of
orbits
and
periodic
orbits,
etc.
we
implicitly
use
the
concept
of
tubular
neighborhoods.
Nonetheless,
these
and
many
other
concepts
and
terminology
are
frequently
used
in
research
literature.
It
is
also
our
intent
to
introduce
the
reader
to
a
more
sophisticated
mathematical
framework
so
that
when
the
reader
ventures
to
research
literature
or
further
reading,
he
will
not
feel
totally
lost
due
to
different
"culture"
or
terminology.
Furthermore,
in
global
theory,
the
state
spaces
may
be
differentiable
manifolds
with
nontrivial
topology.
In
such
cases,
concepts
from
algebraic
topology
are
at
times
essential
to
the
understanding.
In
light
of
the
above
remarks,
for
the
first
reading
the
reader
may
want
to
skip
Section
2.3,
part
of
Section
2.4,
de
Rham
cohomology
part
of
Section
2.8
and
Section
2.9.
The
subjects
discussed
in
Chapter
3
are
not
extensively
utilized
in
the
subsequent
chapters,
at
least
not
explicitly.
Nonetheless,
some
of
the
concepts
and
even
terminology
do
find
their
way
to
our
later
discussions.
This
chapter
is
included,
and
indeed
is
lectured,
to
prepare
the
readers
with
some
concepts
and
understanding
about
global
analysis
in
general,
and
some
techniques
important
to
the
global
theory
of
dynamical
systems.
In
particular,
the
reader
may
find
it
useful
when
they
venture
to
theoretically
oriented
research
literature.
In
the
next
chapter,
we
shall
discuss
the
general
theory
of
dynamical
systems.
Most
of
the
machinary
developed
in
the
last
chapter
is
not
used
immediately.
Only
in
the
last
section
of
Chapter
4,
the
idea
of
linearization
of
nonlinear
ix
differential
operators
will
be
utilized
for
the
discussion
of
linearization
of
dynamical
systems.
Nor
does
Chapter
5
depend
on
the
material
of
Chapter
3.
As
a
consequence,
one
may
want
to
proceed
from
Chapter
2
to
Chapters
4
and
5,
except
Section
6
of
Chapter
4.
Then
come
back
to
Chapter
3
and
continue
to
Chapter
6.
In
Section
4.7
we
briefly
discuss
the
linearization
process
based
on
some
results
from
Chapter
3.
For
most
scientists
and
engineers,
the
linearization
process
is
"trivial".
Everyone
has
done
this
since
their
freshman
year
many
times.
Unfortunately,
the
linearization
process
is
the
most
misunderstood
and
frequently
mistaken
procedure
for
scientists
and
engineers
in
dealing
with
nonlinear
phenomena
and
nonlinear
dynamical
systems
at
large.
This
is
particularly
true
for
highly
nonlinear
systems.
Two
illustrations
from
our
earlier
training
in
mathematics
can
make
the
point
clear.
First,
in
our
freshman
calculus
course,
we
learned
about
the
condition
of
continuity
of
a
function
at
a
point,
and
its
derivative
at
that
point.
For
the
derivative
to
have
a
meaning,
not
only
the
change
of
the
independent
variable,
say,
~x,
has
to
be
very
small
(i.e.,
local),
but
also
the
limits
lim
x
~
O+
~YI
~x
and
lim
x
~
0
_
~y;~x
agree
at
x = x
0
•
In
other
words,
the
usual
linearization
makes
sense,
only
when
it
is
done
locally,
and
any
"displacement"
from
the
point
has
to
be
consistent.
The
consistency
can
be
illustrated
by
the
following
example.
As
we
shall
discuss
in
Chapter
2,
a
differentiable
manifold
M
is
a
topological
manifold
(a
topological
space
with
certain
nice
properties)
endowed
with
a
differentiable
structure.
For
any
two
points
p
and
q
in
M,
A,
B
are
sufficiently
small
neighborhoods
of
p
and
q
in
M
respectively.
At
p
and
q
in
A
and
B,
a
rectangular
coordinate
system
can
be
attached
to
each
of
p
and
q,
and
every
variable
looks
linear
in
A
and
B.
What
makes
the
coordinate
system
or
the
differentiable
structure
go
beyond
the
confine
of
A
or
B
is
that
if
the
intersection
of
A
and
B
is
non-empty,
then
the
coordinate
values
in
A
and
B
have
to
agree
at
all
points
in
the
X
intersection
of
A
and
B.
That
is,
the
differentiable
structure
has
to
be
consistently
and
continuously
agreed
between
A
to
B.
Consequently,
for
a
highly
nonlinear
system
when
one
linearizes
such
a
system,
one
has
to
be
certain
that
the
linearization
procedure
is
"consistent"
in
the
above
sense.
Otherwise,
except
at
or
very
near
the
equilibrium
point(s)
in
the
phase
space,
one
has
no
assurance
as
to
the
correctness
of
the
results
of
linearization.
Related
to
the
above
discussion
of
the
linearization
procedure,
we
would
also
like
to
caution
the
reader
about
numerical
simulation
of
nonlinear
dynamical
systems,
even
when
only
dealing
with
a
local
situation.
First
and
foremost,
do
not
just
code
the
differential
equations
and
let
the
computer
do
the
rest.
One
has
to
analyze
the
characteristics
of
the
system,
namely,
how many
and
what
kind
of
fixed
points,
periodic
orbits,
attractors,
etc.
Then,
let
the
computer
do
the
dirty
work,
and
compare
with
the
analysis
to
~ee
whether
or
not
the
numerical
simulation
agrees
with
the
analysis
on
the
number
and
type
of
characteristics.
Otherwise,
there
is
no
way
one
can
be
sure
the
simulation
is
correct
and
meaningful.
Another
important
point
sometimes
scientists
and
engineers
have
overlooked
is
whether
or
not
the
system
has
a
Cauchy
data
set.
If
the
system
does
not
admit
a
Cauchy
data
set,
it
is
tantamount
to
say
that
the
system
does
not
admit
a
unique
time.
Thus,
the
time
evolution
of
the
system
loses
its
meaning.
One may
argue
that
who
cares
about
t
~
~.
But
even
for
a
finite
t,
the
system
may become
unpredictable,
with
or
without
chaos,
etc.
It
is
also
appropriate
to
point
out
a
practical
point
which
relates
to
the
numerical
schemes
for
studying
the
nonlinear
dynamical
system.
The
implicit
method
is
a
very
popular
and
efficient
scheme
to
study
the
linear
or
weakly
nonlinear
differential
equations.
But
for
highly
nonlinear
systems,
this
scheme
may
leads
to
erroneous
results.
This
is
because
the
interpolation
definitely
introduces
errors,
xi