(6)
k
2
=
1
t
2

t
0
ln
d
2
$ a
1
d
0
=
1
t
2

t
0
ln(a
1
$ a
2
)
k
3
=
1
t
3

t
0
ln
d
3
$ a
2
$ a
1
d
0
=
1
t
3

t
0
ln(a
1
$ a
2
$ a
3
)
®
and so on. The multiplicative factors
are derived in section 3, in
a
1
, a
1
$ a
2
,
¬
case it is not intuitively obvious. We there
fore conclude that the instantaneous Lyapu
nov exponent is
(7)
k
n
=
1
t
n

t
0
S
i
=
1
n
ln a
i
where we have defined
(8)
a
i
h
d
(
t
i
)
d
(
t
0
)
As long as the rescalings take place in the
linear regime, this construction is valid.
Notice that, in a computer, only the accumu
lating sum of the natural log of the
i
need
be stored. In addition, the time intervals
need not be evenly spaced.
2. Renormalization of the Test Orbit.
The rescaling of the test particle orbit is per
formed on the test  reference phase space
distance vector. Whenever the distance d(t)
becomes greater than or equal to the thresh
old D, we scale the test particle distance
from the reference particle by the factor 1/
i
,
maintaining the current relative orientation
between the two particles in phase space.
Write the reference and test particle phase
space vectors as
(9)
R h
x
y
z
v
x
v
y
v
z
ref
and r h
x
y
z
v
x
v
y
v
z
test
Define
. Then the adjustment to
q
q h r

R
the test particle phase space coordinates at
time t
i
is
(10)
r
i
b R
i
+
q
q
i
a
i
Alternatively, one could write the equivalent
expression
(11)
r
i
b r
i

a
i

1
a
i
$ qq
i
Eq. (10) is slightly less expensive to calcu
late than eq. (11). All we are doing is res
caling the distance d(t),
(12)
d
(
t
i
)
b
d
(
t
i
)
a
i
in the appropriate direction in phase space.
I have found that
and
d
0
=
10

6
D
=
10

4