Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
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which
initially
are
small,
but
their
growth
rate
usually
is
also
the
largest.
After
propagating
for
a
short
time,
the
amplitudes
of
the
errors
can
be
as
large
as
the
main
variables.
Small
scale
self-focusing
in
nonlinear
optics
is
a
very
good
example
[Bespalov
and
Talanov
1966;
Fleck
et
al
1976;
Lee
1977;
Brown
1981).
The
two
volume
set
by
R.
Bellman
(Methods
of
Nonlinear
Analysis,
Academic
Press,
1973}
provides
eloquent
motivation
and
many
techniques
particularly
useful
for
numerical
simulation
of
nonlinear
systems.
This
set
is
highly
recommended
for
anyone
seriously
interested
in
the
numerical
simulation
of
nonlinear
dynamical
systems.
In
Chapter
5
we
introduce
the
Liapunov's
direct
(or
second)
method
for
analyzing
stability
properties
of
nonlinear
dynamnical
systems.
In
this
chapter,
we
are
still
dealing
with
the
local
theory
of
stability
for
nonlinear
dynamical
systems.
The
first
half
of
Chapter
6
discusses
the
global
theory
of
stability
and
the
very
important
concept
of
structural
stability.
Sections
6.6
on
bifurcations
and
6.7
on
chaos
could
be
grouped
together
with
the
local
theory
of
stability.
Nonetheless,
because
some
of
the
concepts
and
techniques
are
also
useful
for
the
global
theory,
we
put
these
two
sections
in
this
chapter
and
deal
with
global
theory
concerned
with
bifurcations
and
chaos.
A
recent
new
definition
of
stability
proposed
by
Zeeman,
which
is
intended
to
replace
the
original
structural
stability,
is
very
interesting
and
has
a
great
potential
in
analyzing
the
global
stability
of
many
practical
physics
problems.
We
did
not
include
any
discussion
on
quantum
chaos,
because
this
author
does
not
understand
it.
We
only
mention
fractals
in
passing,
not
that
it
is
unimportant,
(on
the
contrary
it
is
a
very
important
topic),
but
mainly
due
to
lack
of
space.
Chapter
7
discusses
applications
of
stability
analyses
to
various
disciplines,
and
we
also
indicate
the
commonality
and
the
similarities
of
the
mathematical
structures
for
various
problems
in
diverse
disciplines.
For
instance,
in
Section
7.5,
we
not
only
discuss
the
dynamical
processes
of
competitive
interacting
population
processes
in
biology
and
population
ecology
and
biochemical
autocatalysis
processes,
but
we
have
also
pointed
out
that
these
equations
also
describe
some
processes
in
laser
physics
and
semiconductor
physics,
to
name a
few.
As
another
example,
the
discussion
of
permanence
in
Section
7.5
with
minor
modifications,
can
also
be
applied
to
mode
structures
and
phased
arrays
in
lasers.
The
point
is
that
scientists
and
engineers
can
learn,
or
even
directly
apply
results,
from
different
disciplines
to
solve
their
problems.
Putting
it
differently,
physical
scientists
can
learn
from,
and
should
communicate
with,
biological
scientists,
and
vice
versa.
One way
for
the
physical
scientists
to
establish
such
communication
is
to
read
some
biological
journals.
The
conventional
wisdom
has
it
that
papers
in
biological
journals
are
not
very
sophisticated
mathematically.
The
fact
is,
in
the
past
decade
or
so,
there
has
been
an
influx
of
mathematicians
working
on
mathematical
biology.
Consequently,
the
whole
landscape
of
the
theoretical
or
mathematical
biology
journals
has
changed
dramatically.
Nowadays,
there
are
many
deep
and
far
reaching
mathematical
papers
in
a much
broader
context
being
published
in
those
journals.
The
second
goal
of
this
volume
is
to
draw
the
attention
for
such
"lateral"
interactions
between
physical
and
biological
scientists.
The
third
goal
is
to
provide
the
reader
a
very
personal
guide
to
study
the
global
nonlinear
dynamical
systems.
We
provide
the
concepts
and
methods
of
analyzing
problems,
but
we
do
not
provide
"recipes".
After
all,
it
is
not
intended
to
be
a
''cookbook".
Should
a
cookbook
be
contemplated,
the
"menu"
would
be
very
limited.
We
have
tried
to
include
some
of
the
references
the
author
has
benefited
from.
surely
they
are
far
from
comprehensive,
and
the
list
is
very
subjective
to
say
the
least.
The
omission
of
some
of
the
important
works
indicates
the
ignorance
of
the
author.
xiii
The
following
books
are
highly
recommended
for
undergraduate
or
beginners
on
dynamical
systems:
Hirsch
and
Smale
[1974],
Iooss
and
Joseph
[1980],
Irwin
[1980],
Ruelle
[1989],
Thompson
and
Stewart
[1986].
Without
inspiring
teachers
like
Peter
Bergmann,
Heinz
Helfenstein,
John
Klauder,
and
Douglas
Anderson,
I
would
not
have
the
opportunity
to
learn
as
much.
My
colleagues
and
friends,
David
Brown,
Ying-Chih
Chen,
Da-Wen
Chen,
Gerrit
Smith,
and
many
others,
are
also
my
teachers.
They
not
only
have
taught
me
various
subjects,
but
they
also
provided
me
with
enjoyable
learning
experiences
throughout
my
professional
career.
I
would
also
like
to
acknowledge
the
understanding
and
support
of
my
two
childern,
Jennifer
and
Peter,
without
their
encouragement
I
would
not
be
able
to
complete
this
project.
It
is
my
pleasure
to
acknowledge
the
constant
encouragement,
support,
suggestions
for
improvements,
and
patience,
of
the
editorial
staff
of
the
publisher,
in
particular,
Mr.
K.
L.
Choy.
Kotik
K.
Lee
Colorado
Springs
CONTENTS
Preface
Chapter
1
Introduction
1.
1
What
is
a
dynamical
system?
1.2
What
is
stability,
and
why
should
we
care
about
it?
Chapter
2
Topics
in
Topology
and
Differential
Geometry
2.1
Getting
to
the
basics
-
algebra
2.
2
Bird'
s
eye
view
of
general
topology
2.
3
Algebraic
topology
2.4
Elementary
differential
topology
and
differential
geometry
2.5
Critical
points,
Morse
theory,
and
transversality
2.
6
Group
and
group
actions
on
manifolds,
Lie
groups
2.
7
Fiber
bundles
2.
8
Differential
forms
and
exterior
algebra
2.
9
Vector
bundles
and
tubular
neighborhoods
Chapter
3
Introduction
to
Global
Analysis
and
Infinite
Dimensional
Manifolds
3.1
What
is
global
analysis?
3.
2
Jet
bundles
3.
3
Whitney
C"
topology
3.
4
Infinite
dimensional
manifolds
3.
5
Differential
operators
Chaper
4
General
Theory
of
Dynamical
Systems
4.
1
Introduction
4.
2
Equivalence
relations
4.3
Limit
sets
and
non-wandering
sets
XV
vii
1
1
18
24
25
27
40
45
57
73
82
92
102
111
111
112
118
122
128
145
145
152
156
4.4
Velocity
fields,
integrals,
and
ordinary
differential
equations
4.
5
Dispersive
systems
4.
6
Linear
systems
4.
7
Linearization
Chapter
5
Stability
Theory
and
Liapunov'
s
Direct
Method
5.1
Introduction
5.2
Asymptotic
stability
and
Liapunov's
theorem
5.
3
Converse
theorems
5.
4
Comparison
methods
5.5
Total
stability
5.6
Popov's
frequency
method
to
construct
a
Liapunov
function
168
174
178
187
198
198
206
221
222
225
229
5.7
Some
topological
properties
of
regions
of
attraction
231
5.
8
Almost
periodic
motions
238
Chapter
6
Introduction
to
the
General
Theory
of
Structural
Stability
6.1
Introduction
6.2
Stable
manifolds
of
diffeomorphisms
and
flows
6.3
Low
dimensional
stable
systems
6.
4
Anosov
systems
6.5
Characterizing
structural
stability
6.
6
Bifurcation
6.
7
Chaos
6.
8 A
new
definition
of
stability
Chapter
Applications
7.
1
Introduction
xvi
241
241
246
252
260
260
267
280
314
324
324
7.2
Damped
oscillators
and
simple
laser
theory
7.3
Optical
instabilities
7.4
Chemical
reaction-diffusion
equations
7.5
Competitive
interacting
populations,
autocatalysis,
and
permanence
7.6
Examples
in
semiconductor
physics
and
semiconductor
lasers
7.7
Control
systems
with
delayed
feedback
7.8
Semiconductor
laser
linewidth
reduction
by
feedback
control
and
phased
arrays
References
Index
xvii
327
341
351
359
375
379
385
399
443
Chapter
1
Introduction
1.1
What
is
a
dynamical
system?
A
dynamical
system
can
be
thought
of
as
any
set
of
equations
giving
the
time
evolution
of
the
state
of
the
system
from
the
knowledge
of
its
previous
history.
Nearly
all
observed
phenomena
in
scientific
investigation
or
in
our
daily
lives
have
important
dynamical
aspects.
Examples
are:
(a)
in
physical
sciences:
Newton's
equations
of
motion
for
a
particle
with
suitably
specified
forces,
Maxwell's
equations
for
electrodynamics,
Navier-Stokes
equations
for
fluid
motions,
time-dependent
Schrodinger's
equation
in
quantum
mechanics,
and
chemical
kinetics:
(b)
in
life
systems:
genetic
transference,
embryology,
ecological
decay,
and
population
growth:
(c)
and
in
social
systems:
economical
structure,
the
arms
race,
or
promotion
within
an
organizational
hierarchy.
Although
these
examples
illustrate
the
pervasiveness
of
dynamic
situations
and
the
potential
value
of
developing
the
facility
for
modeling
(representing)
and
analyzing
the
dynamic
behavior,
it
should
be
emphasized
that
the
general
concept
of
dynamics
and
the
treatment
of
dynamical
systems
transcends
the
particular
origin
or
the
setting
of
the
processes.
In
our
daily
lives
we
often
quite
effectively
deal
with
many
simple
dynamic
situations
which
can
be
understood
and
analyzed
intuitively
(i.e.,
by
experience)
without
resorting
to
mathematics
and
the
general
theory
of
dynamical
systems.
Nonetheless,
in
order
to
approach
complex
and
unfamiliar
situations
efficiently,
it
is
necessary
to
proceed
systematically.
Mathematics
can
provide
the
required
conceptual
framework
and
proper
language
to
analyze
such
complex
and
unfamiliar
dynamic
situations.
In
view
of
its
mathematical
structure,
the
term
dynamics
takes
on
a
dual
meaning.
First,
as
stated
earlier,
it
is
a
term
for
the
time-evolutionary
phenomena
around
us
and
about
us:
and
second,
it
is
a
term
for
the
pact
of
mathematics
1
which
is
used
to
represent
and
analyze
such
phenomena,
and
the
interplay
between
both
aspects.
Although
there
are
numerous
examples
of
interesting
dynamic
situations_arising
in
various
areas,
the
number
of
corresponding
general
forms
for
mathematical
representation
is
limited.
Most
commonly,
dynamical
systems
are
represented
mathematically
in
terms
of
either
differential
or
difference
equations.
In
fact,
in
terms
of
the
mathematical
content,
the
elementary
study
of
dynamics
is
almost
synonymous
with
the
theory
of
differential
and
difference
equations.
Before
proceeding
to
the
quantitative
description
of
dynamical
systems,
one
should
note
that
there
are
qualitative
structures
of
dynamical
systems
which
are
of
fundamental
importance,
as
will
be
discussed
later.
At
the
moment,
it
is
suffice
to
note
that
even
though
there
are
many
different
disciplines
in
the
natural
sciences,
let
alone
many
more
subfields
in
each
discipline,
Nature
seems
to
follow
the
economical
principle
that
a
tremendous
number
of
results
can
be
condensed
into
a few
simple
laws
which
summarize
our
knowledge.
These
laws
are
qualitative
in
nature.
It
should
be
emphasized
that
here
qualitative
does
not
mean
poorly
quantitative,
rather
topologically
invariant,
i.e.,
independent
of
local
and
detail
descriptions.
Furthermore,
common
to
many
natural
phenomena,
besides
their
qualitative
similarity,
is
their
universality
where
the
details
of
the
interactions
of
systems
undergoing
spontaneous
transitions
are
often
irrelevent.
This
calls
for
topological
descriptions
of
the
phenomena
under
consideration.
Hence,
the
concept
of
structural
stability
and
the
theories
of
singularity
and
bifurcation
undoubtedly
lead
the
way.
Simply
stated,
the
use
of
either
differential
or
difference
equations
to
represent
dynamic
behavior
corresponds
to
whether
the
behavior
is
viewed
as
occurring
in
continuous
or
discrete
time
respectively.
Continuous
time
corresponds
to
our
usual
perception
that
time
is
often
2
viewed
as
flowing
smoothly
past
us.
In
mathematical
terms,
continuous
time
is
quantified
by
the
continuum
of
real
numbers
and
usually
denoted
by
the
parameter
t.
Dynamic
behavior
viewed
in
continuous
time
is
usually
described
by
differential
equations.
Discrete
time
consists
of
an
ordered
set
rather
than
a
continuous
parameter
represented
by
real
numbers.
Usually
it
is
convenient
to
introduce
discrete
time
when
events
occur
or
are
accounted
for
only
at
discrete
time
periods.
For
instance,
when
developing
a
population
model,
it
may
be
convenient
to
work
with
annual
population
changes,
and
the
data
is
normally
available
annually,
rather
than
continually.
Discrete
time
is
usually
labeled
by
simple
indexing
of
variables
in
order
and
starting
at
a
convenient
reference
point.
Thus,
dynamic
behavior
in
discrete
time
is
usually
described
by
equations
relating
the
value
of
a
variable
at
one
time
to
the
values
of
variables
at
adjacent
times.
such
equations
are
called
difference
equations.
Furthermore,
in
order
to
calculate
the
dynamics
of
a
system,
which
are
normally
represented
by
differential
equations,
with
an
infinite
degree
of
freedom
(we
shall
come
to
this
shortly)
such
as
fluids,
it
is
more
convenient
to
numerically
break
down
the
system
into
small
but
finite
cells
in
space
and
discrete
periods
of
time.
Thus
one
also
uses
the
difference
method
to
solve
differential
equations.
In
what
follows,
we
shall
concentrate
on
the
aspect
of
continuous
time,
i.e.,
the
differential
equations
aspect
of
dynamical
systems.
In
the
late
1800's,
Henri
Poincare
initiated
the
qualitative
theory
of
ordinary
differential
equations
in
his
famous
menoir
[1881,
1882].
Ever
since
then,
differential
topology,
a
modern
development
of
calculus,
has
provided
the
proper
setting
for
this
qualitative
theory.
As
we
know,
ordinary
differential
equations
appear
in
many
different
disciplines,
and
the
qualitative
theory
often
gives
some
important
insight
into
the
physical,
biological,
or
social
realities
of
the
situations
studied.
And
the
qualitative
3