138
5 Arnol’d Diffusion
resonance conditions
j n j ω j ({I j }) = 0, where ω j = ∂H o /∂I j and n j range over
all integers. For small , each resonance line has a width of order (or smaller than)
√
(the width of a pendulum separatrix—see Appendix B) and a stochastic layer
associated with it. Trajectories can diffuse along these stochastic layers.
Nekhoroshev showed that, for very small , diffusion along the stochastic layers
may take a very long time (Nekhoroshev 1971, 1977). He found that, if I j (0) is the
initial value of an action variable then, in time τ N , it will have traveled a “distance”
|I j (τ N )−I j (0)|∼ β along a stochastic layer in a time of order τ N ∼(1//)exp(1// α ),
where 0 < α < 1 and 0 < β < 1. For small , this is indeed a very long time.
As increases, the width of the resonance regions increases until the system
reaches the Chirikov regime (Chirikov 1979), where resonances not only cross but
also begin to overlap. Then, chaotic regions of the phase space and random diffusion
through the phase space can become global.
One of the systems originally considered by Arnold (1963) in describing
diffusion in the phase space of systems with three or more DoF had the Hamiltonian
(the Arnol’d Hamiltonian)
H =
1
2
(J
2
1 + J
2
2 ) + (cos(θ 1 ) − 1)(1 + μ sin(θ 2 ) + μ cos(t)),
(5.9)
where (J 1 , J 2 , θ 1 , θ 2 ) are action-angle variables and and μ are small coupling
parameters. This system contains six primary resonances. If we introduce the
coordinates (p, x = t), we can write Eq. (5.9) in the time-independent form
H =
1
2
(J
2
1 + J
2
2 ) + p + (cos(θ 1 ) − 1) − μμ cos(x) − μμ sin(θ 2 )
+
μμ
2
[sin(θ 2 − θ 1 ) + sin(θ 2 + θ 1 ) + cos(θ 1 − x) + cos(θ 1 + x)]. (5.10)
The locations of the six primary resonances are determined by the equations
˙
θ 1 ≈ J 1 = 0, ˙
θ 2 ≈ J 2 = 0, ˙
θ 1 ± ˙
θ 2 ≈ J 1 ± J 2 = 0, ˙
θ 1 ± ˙
x ≈ J 1 ± 1 = 0.
(5.11)
The resonance zone, J 1 = 0, has a width proportional to
√
, while all other
primary resonance zones have a width proportional to
√ μ. For and small μ,
the partial energy surface, H o =
1
2 (J 2
1 + J 2
2 ) + p, is the same as that in Fig. 5.1.
The resonance zones are given approximately by the intersection of the resonance
surfaces in Eq. (5.11) with the unperturbed partial energy surface. The projection
of the resonance curves onto the J 1 − J 2 plane and the respective widths of the
resonance zones are shown in Fig. 5.3. Note that for a trajectory starting in the
stochastic layer of the resonance zone centered at J 1 = 0, the change in J 1 due
to diffusion across the resonance zone (in the ±J 1 direction) is constrained to the
width of the resonance. However, the value of J 2 can increase significantly due to
diffusion along the J 1 = 0 resonance (in the ±J 2 direction). A trajectory can also
escape the J 1 = 0 resonance by moving onto one of the smaller resonances that
5 Arnol’d Diffusion
resonance conditions
j n j ω j ({I j }) = 0, where ω j = ∂H o /∂I j and n j range over
all integers. For small , each resonance line has a width of order (or smaller than)
√
(the width of a pendulum separatrix—see Appendix B) and a stochastic layer
associated with it. Trajectories can diffuse along these stochastic layers.
Nekhoroshev showed that, for very small , diffusion along the stochastic layers
may take a very long time (Nekhoroshev 1971, 1977). He found that, if I j (0) is the
initial value of an action variable then, in time τ N , it will have traveled a “distance”
|I j (τ N )−I j (0)|∼ β along a stochastic layer in a time of order τ N ∼(1//)exp(1// α ),
where 0 < α < 1 and 0 < β < 1. For small , this is indeed a very long time.
As increases, the width of the resonance regions increases until the system
reaches the Chirikov regime (Chirikov 1979), where resonances not only cross but
also begin to overlap. Then, chaotic regions of the phase space and random diffusion
through the phase space can become global.
One of the systems originally considered by Arnold (1963) in describing
diffusion in the phase space of systems with three or more DoF had the Hamiltonian
(the Arnol’d Hamiltonian)
H =
1
2
(J
2
1 + J
2
2 ) + (cos(θ 1 ) − 1)(1 + μ sin(θ 2 ) + μ cos(t)),
(5.9)
where (J 1 , J 2 , θ 1 , θ 2 ) are action-angle variables and and μ are small coupling
parameters. This system contains six primary resonances. If we introduce the
coordinates (p, x = t), we can write Eq. (5.9) in the time-independent form
H =
1
2
(J
2
1 + J
2
2 ) + p + (cos(θ 1 ) − 1) − μμ cos(x) − μμ sin(θ 2 )
+
μμ
2
[sin(θ 2 − θ 1 ) + sin(θ 2 + θ 1 ) + cos(θ 1 − x) + cos(θ 1 + x)]. (5.10)
The locations of the six primary resonances are determined by the equations
˙
θ 1 ≈ J 1 = 0, ˙
θ 2 ≈ J 2 = 0, ˙
θ 1 ± ˙
θ 2 ≈ J 1 ± J 2 = 0, ˙
θ 1 ± ˙
x ≈ J 1 ± 1 = 0.
(5.11)
The resonance zone, J 1 = 0, has a width proportional to
√
, while all other
primary resonance zones have a width proportional to
√ μ. For and small μ,
the partial energy surface, H o =
1
2 (J 2
1 + J 2
2 ) + p, is the same as that in Fig. 5.1.
The resonance zones are given approximately by the intersection of the resonance
surfaces in Eq. (5.11) with the unperturbed partial energy surface. The projection
of the resonance curves onto the J 1 − J 2 plane and the respective widths of the
resonance zones are shown in Fig. 5.3. Note that for a trajectory starting in the
stochastic layer of the resonance zone centered at J 1 = 0, the change in J 1 due
to diffusion across the resonance zone (in the ±J 1 direction) is constrained to the
width of the resonance. However, the value of J 2 can increase significantly due to
diffusion along the J 1 = 0 resonance (in the ±J 2 direction). A trajectory can also
escape the J 1 = 0 resonance by moving onto one of the smaller resonances that