2.5 Nonlinear Resonance and Chaos
27
and
dJ 2
dt
= 2αJ 2 sin(2 2 )(I
− J 2 ),
(2.43)
dd 2
dt
= −I
+ 6J 2 + α cos(2 2 )(I
− 2J 2 ).
(2.44)
Since J 1 is constant, Eqs. (2.43) and (2.44) can be solved first for J 2 (t) and 2 (t)
and then substituted into Eq. (2.42) to obtain 1 (t).
Let us now find the fixed points of these equations. The fixed points are points
for which
dJ 2
dt = 0 and
dd 2
dt = 0. Fixed points occur when 2 =
nπ
2 and J 2 = J o ,
where J o is a solution of the equation
− I
+ 6J o + α cos(nπ )(I
− 2J o ) = 0.
(2.45)
Note that for α 1, J o ≈
I
6 .
The nature of the fixed points can be determined by linearizing the equations
of motion about points (J 2 = J o , , 2 =
nπ
2 ). We let J 2 (t) = J o + J(t) and
2 (t) =
nπ
2 + (t) and linearize in J(t) and We find
d
dt
J(t)
(t)
=
0
4 α cos(nπ )J o (I − J o )
(6 − 2α cos(nπ ))
0
J(t)
(t)
. (2.46)
The solution
J(t)
(t)
to Eq. (2.46) determines the manner in which trajectories flow
in the neighborhood of the fixed points. For α 1 (and therefore J o ≈
I
6 ), these
equations reduce to
d
dt
J(t)
(t)
≈
0
20αI 2
36 cos(nπ )
6
0
J(t)
(t)
.
(2.47)
Let us assume that Eq. (2.47) has a solution of the form
J(t)
= e
λt
A J
A
,
(2.48)
where A J and A are independent of time. Then we can solve the resulting
eigenvalue equation
λ
A J
A
=
0
20αI 2
36 cos(nπ )
6
0
A J
A
for both λ and
A J
A
. The eigenvalues are given by