2.5 Nonlinear Resonance and Chaos
29
Fig. 2.6 Phase space
trajectories for the (2,2)
resonance Hamiltonian in
Eq. (2.40) (p = −(2J 2 )
1
2
sin(( 2 ) and q = (2J 2 )
1
2
cos(( 2 )). For all curves,
E = 0.18 and α = 0.1. The
curves consist of discrete
points because we have
plotted points along the
trajectories at discrete times
and
J 2 (t) = I 2 +
2αI 1 I 2 cos(2ω 1 t − 2ω 2 t)
(2ω 1 − 2ω 2 )
.
(2.52)
In order for these equations to have meaning, the following condition must hold:
|2ω 1 − 2ω 2 | = |2I 1 − 10I 2 | | 2αI 1 I 2 .
However, near a resonance, I 1 ≈ 5I 2 . Therefore this condition breaks down in
the neighborhood of a resonance zone. Actually this is to be expected since the
resonance introduces a topological change in the flow pattern in the phase space.
2.5.1.2 (2,3) Resonance
Walker and Ford also studied a (2,3) resonance with Hamiltonian
H = H o (J 1 , J 2 ) + βJ 1 J
3
2
2 cos(2θ 1 − 3θ 2 ) = E.
(2.53)
This again is integrable and has two isolating integrals of the motion, the Hamiltonian, H , and
I = 3J 1 + 2J 2 = C 3 .
(2.54)