3.8 Renormalization Map
85
nances outside this region tend to average out. Therefore, in order to analyze the
mechanism for the destruction of KAM tori between two primary resonances, it is
often sufficient to consider a Hamiltonian composed of only those two resonances,
and that is the basis of the renormalization theory we describe below.
Escande and Doveil (1981); Escande (1982, 1985) built a renormalization theory
based on a system with two primary resonances. The Hamiltonian for the two
resonance system, which they call the paradigm Hamiltonian, has the form
H (α) =
p 2
α
2
− U
(0)
α cos(x α ) − U
(1)
α cos[ν α (x α − t α )],
(3.71)
and depends on only three parameters: the amplitudes of two cosine waves, U
(0)
α
and U
(1)
α , and the relative wave number of the two waves ν α =
N α
M α
(M α and N α are
relatively prime integers). The index, α, is the iteration step of the renormalization
map. This Hamiltonian has period 2πM α . The relative wave number, ν α , is the
relative number of oscillations of the two cosine waves during this period. The
resonance that results from the cosine wave with amplitude U
(0)
α has speed ˙
x α = 0
and half-width p α = 2
U
(0)
α , while the resonance that results from the cosine
wave with amplitude U
(1)
α has speed ˙
x α = 1 and half-width p α = 2
U
(1)
α . From
the resonance condition ˙
x α =
∂H
∂p α
≈ p α , these resonances are located at p α = 0
and p α = 1, respectively.
It is possible to use the Chirikov overlap criterion to obtain a “back of the
envelope” estimate of the parameter values at which two neighboring resonances
overlap. This criterion works fairly well as long as the resonances have pendulumlike structure in the surface of section. The Chirikov overlap criterion says that
overlap occurs when the separatrices of the two resonance structures touch. When
this happens, the last KAM torus between the two resonances is destroyed. Thus,
overlap occurs when
S = 2
U
(0)
α + 2
U
(1)
α = 1.
(3.72)
Strobe plots obtained from a numerical solution of Hamilton’s equations for the
system described by the Hamiltonian in Eq. (3.71), for the case ν α = 1 and U
(0)
α =
U
(1)
α , are shown in Figs. 3.24 and 3.25.
In Fig. 3.24a, we are well below the Chirikov estimate for overlap and indeed
the primary resonances are well-separated. Secondary islands and KAM tori are
clearly shown in this plot. In Fig. 3.24b, we are still well below the Chirikov estimate
for overlap. However, the last KAM torus has been destroyed, and overlap has
clearly occurred in this plot. There is a chaotic trajectory that extends from the
neighborhood of one resonance to the neighborhood of the other. However, there is
still a lot of structure in the chaotic sea. In Fig. 3.25, we are at the value of S where
the simple Chirikov overlap criterion predicts overlap. However, there is already a
85
nances outside this region tend to average out. Therefore, in order to analyze the
mechanism for the destruction of KAM tori between two primary resonances, it is
often sufficient to consider a Hamiltonian composed of only those two resonances,
and that is the basis of the renormalization theory we describe below.
Escande and Doveil (1981); Escande (1982, 1985) built a renormalization theory
based on a system with two primary resonances. The Hamiltonian for the two
resonance system, which they call the paradigm Hamiltonian, has the form
H (α) =
p 2
α
2
− U
(0)
α cos(x α ) − U
(1)
α cos[ν α (x α − t α )],
(3.71)
and depends on only three parameters: the amplitudes of two cosine waves, U
(0)
α
and U
(1)
α , and the relative wave number of the two waves ν α =
N α
M α
(M α and N α are
relatively prime integers). The index, α, is the iteration step of the renormalization
map. This Hamiltonian has period 2πM α . The relative wave number, ν α , is the
relative number of oscillations of the two cosine waves during this period. The
resonance that results from the cosine wave with amplitude U
(0)
α has speed ˙
x α = 0
and half-width p α = 2
U
(0)
α , while the resonance that results from the cosine
wave with amplitude U
(1)
α has speed ˙
x α = 1 and half-width p α = 2
U
(1)
α . From
the resonance condition ˙
x α =
∂H
∂p α
≈ p α , these resonances are located at p α = 0
and p α = 1, respectively.
It is possible to use the Chirikov overlap criterion to obtain a “back of the
envelope” estimate of the parameter values at which two neighboring resonances
overlap. This criterion works fairly well as long as the resonances have pendulumlike structure in the surface of section. The Chirikov overlap criterion says that
overlap occurs when the separatrices of the two resonance structures touch. When
this happens, the last KAM torus between the two resonances is destroyed. Thus,
overlap occurs when
S = 2
U
(0)
α + 2
U
(1)
α = 1.
(3.72)
Strobe plots obtained from a numerical solution of Hamilton’s equations for the
system described by the Hamiltonian in Eq. (3.71), for the case ν α = 1 and U
(0)
α =
U
(1)
α , are shown in Figs. 3.24 and 3.25.
In Fig. 3.24a, we are well below the Chirikov estimate for overlap and indeed
the primary resonances are well-separated. Secondary islands and KAM tori are
clearly shown in this plot. In Fig. 3.24b, we are still well below the Chirikov estimate
for overlap. However, the last KAM torus has been destroyed, and overlap has
clearly occurred in this plot. There is a chaotic trajectory that extends from the
neighborhood of one resonance to the neighborhood of the other. However, there is
still a lot of structure in the chaotic sea. In Fig. 3.25, we are at the value of S where
the simple Chirikov overlap criterion predicts overlap. However, there is already a