64
3 Area-Preserving Maps
These are the best rational approximates to ω in the sense of a Diophantine
approximation. That is,
ω −
N
M
>
ω −
N i
M i
for all other
N
M with M < M i+1 (Hardy and Wright 1979). Finite-length
continued fractions are unique up to an ambiguity in the last partial quotient,
[a 0 , a 1 , . . . , a N ] = [a 0 , a 1 , . . . , a N −1 , 1]. For winding numbers 0 ≤ ω ≤ 1, we
must have a 0 = 0 and a i ≥ 1.
The most irrational number is given by the sequence a i = 1 for all i. That is,
[1, 1, 1, . . . , 1, . . .] ≡ γ =
(1 +
√
(5))
2
.
(3.27)
The number γ is called the golden mean and is considered to be the most irrational
number because it is hardest to approximate by rationals (Prasad 1948; MacKay
1983a). In fact, all the sequences ending in a series of 1’s are the slowest-converging
sequences and represent those irrational numbers that are the hardest to approximate
by rational numbers. This has important consequences for dynamics.
We know that KAM tori are destroyed by resonances between degrees of freedom
whose periods are rationally related. Thus, each rational approximate will be
associated to a resonance region in the phase space and a corresponding island chain.
As the parameter K increases, these resonance regions grow and finally destroy
their neighboring KAM tori. However, those KAM tori whose winding numbers are
approximated by sequences of the form
ω = [a 0 , a 1 , . . . , a i , 1, 1, . . . , 1, . . .],
(3.28)
where all entries after a i are 1’s, will be the most irrational and the hardest to
approximate by rational numbers. KAM tori with winding numbers of the type given
in Eq. (3.28) are called noble KAM tori.
In the standard map shown in Figs. 3.10, 3.11, and 3.12, the last KAM tori to be
destroyed are those with winding numbers
ω = [0, 1, 1, 1, . . . , 1, . . .] =
1
γ
≈ 0.618034
(3.29)
and
ω = [0, 2, 1, 1, . . . , 1, . . .] =
1
γ
2
≈ 0.381966.
(3.30)
These noble KAM tori are easily seen in Fig. 3.11d. The rational approximates can
also be seen in these maps. Let us consider the noble KAM torus with winding
3 Area-Preserving Maps
These are the best rational approximates to ω in the sense of a Diophantine
approximation. That is,
ω −
N
M
>
ω −
N i
M i
for all other
N
M with M < M i+1 (Hardy and Wright 1979). Finite-length
continued fractions are unique up to an ambiguity in the last partial quotient,
[a 0 , a 1 , . . . , a N ] = [a 0 , a 1 , . . . , a N −1 , 1]. For winding numbers 0 ≤ ω ≤ 1, we
must have a 0 = 0 and a i ≥ 1.
The most irrational number is given by the sequence a i = 1 for all i. That is,
[1, 1, 1, . . . , 1, . . .] ≡ γ =
(1 +
√
(5))
2
.
(3.27)
The number γ is called the golden mean and is considered to be the most irrational
number because it is hardest to approximate by rationals (Prasad 1948; MacKay
1983a). In fact, all the sequences ending in a series of 1’s are the slowest-converging
sequences and represent those irrational numbers that are the hardest to approximate
by rational numbers. This has important consequences for dynamics.
We know that KAM tori are destroyed by resonances between degrees of freedom
whose periods are rationally related. Thus, each rational approximate will be
associated to a resonance region in the phase space and a corresponding island chain.
As the parameter K increases, these resonance regions grow and finally destroy
their neighboring KAM tori. However, those KAM tori whose winding numbers are
approximated by sequences of the form
ω = [a 0 , a 1 , . . . , a i , 1, 1, . . . , 1, . . .],
(3.28)
where all entries after a i are 1’s, will be the most irrational and the hardest to
approximate by rational numbers. KAM tori with winding numbers of the type given
in Eq. (3.28) are called noble KAM tori.
In the standard map shown in Figs. 3.10, 3.11, and 3.12, the last KAM tori to be
destroyed are those with winding numbers
ω = [0, 1, 1, 1, . . . , 1, . . .] =
1
γ
≈ 0.618034
(3.29)
and
ω = [0, 2, 1, 1, . . . , 1, . . .] =
1
γ
2
≈ 0.381966.
(3.30)
These noble KAM tori are easily seen in Fig. 3.11d. The rational approximates can
also be seen in these maps. Let us consider the noble KAM torus with winding