Astronomical Applications Department, U.S. Naval Observatory lyapcalc Page 3
(13)
x
2
+
y
2
+
z
2
+
v
x
2
+
v
y
2
+
v
z
2
and using only the configuration space
distance
(14)
x
2
+
y
2
+
z
2
is indiscernible.
3. Explanation of summation
1
,
(15)
d
1
=
d
0
e
k
1
t
1
Upon rescaling,
(16)
d
1
b
d
0
e
k
1
t
1
a
1
At the next rescaling time t
2
,
(17)
d
2
=
d
1
e
k
1
$(t
2
-
t
1
)
where d
1
is the rescaled value (i.e., d
0
).
Inserting eq. (16) for d
1
, we have
(18)
d
2
=
d
0
e
k
2
t
2
a
1
f:\dynamics\text\lyapcalc.lwp
Maximum Lyapunov Exponent Calculation
7:07pm September 30, 1995
3
10
2
10
3
10
4
t (orbits)
10
-2
10
-1
10
0
(orbits
-
1
)
threshold = 10
-4
Time: 9:34:34, Date: Jul 9, 1995, Sun
10
-15
10
-12
10
-13
10
-14
10
-11
10
-10
10
-7
10
-8
10
-9
d
0
=10
-5
, 10
-6
