22
1. Basic Theory of Dynamical Systems
Consider a general physical system that is capable of assuming various
“states” described by points in a set
Z
. For example,
Z
might be
R
3
×
R
3
and a state might be the position and momentum (
q
,
p
) of a particle. As
time passes, the state evolves. If the state is
z
0
∈
Z
at time
t
0
and this
changes to
z
at a later time
t
, we set
F
t,t
0
(
z
0
)=
z
and call
F
t,t
0
the
evolution operator
; it maps a state at time
t
0
to what
the state would be at time
t
; i.e., after time
t

t
0
has elapsed. “Determin
ism” is expressed by the law
F
t
2
,t
1
◦
F
t
1
,t
0
=
F
t
2
,t
0
F
t,t
= identity
,
sometimes called the
ChapmanKolmogorov law
.
The evolution laws are called
time independent
when
F
t,t
0
depends
only on
t

t
0
; i.e.,
F
t,t
0
=
F
s,s
0
if
t

t
0
=
s

s
0
.
Setting
F
t
=
F
t,
0
,
the preceding law becomes the
group property
:
F
τ
◦
F
t
=
F
τ
+
t
,F
0
= identity
.
We call such an
F
t
a
fow
and
F
t,t
0
a
timedependent fow
, or an evo
lution operator. If the system is de±ned only for
t
≥
0, we speak of a
semifow
.
It is usually not
F
t,t
0
that is given, but rather the
laws oF motion
. In
other words, di²erential equations are given that we must solve to ±nd the
³ow. In general,
Z
is a manifold (a generalization of a smooth surface), but
we con±ne ourselves here to the case that
Z
=
U
is an open set in some
Euclidean space
R
n
. These equations of motion have the form
dx
dt
=
X
(
x
)
,x
(0) =
x
0
where
X
is a (possibly timedependent) vector ±eld on
U
.
Example.
The motion of a particle of mass
m
under the in³uence of the
gravitational force ±eld is determined by Newton’s second law:
m
d
2
r
dt
2
=
F
;
i.e., by the ordinary di²erentatial equations
m
d
2
x
dt
2
=

mMGx
r
3
;
m
d
2
y
dt
2
=

mMGy
r
3
;
m
d
2
z
dt
2
=

mMGz
r
3
;