3.5 Renormalization in Twist Maps
69
3.5.2 Nonintegrable Twist Map
The procedure described in Sect. 3.5.1 can be extended to nonintegrable maps such
as the standard map for K =0. Shenker and Kadanoff (1982) showed that the rational
approximates exhibit self-similarity when K = K ∗ = 0.9716354. MacKay (1982,
1983a) showed that a set of mappings like those of Eqs. (3.41) and (3.42) could
be constructed for K =0 but with different spatial scaling, B i . MacKay also found
that the fixed point Eqs. (3.43) and (3.44) appear to have another solution, which he
called the critical fixed point, which determines the behavior of the neighborhood
of the inverse golden mean torus for the critical parameter K = K ∗ .
MacKay (1982) determined the critical parameter, K ∗ , by finding the parameter
K = K ∗
i at which the ith dominant elliptic fixed point becomes unstable. In the
limit i → ∞, he finds
lim
i→∞
K
∗
i → K
∗
= 0.9716354,
(3.49)
while the rate of approach to the critical value is given by
δ = lim
i→∞
K ∗
i − K ∗
i−1
K ∗
i+1 − K ∗
i
≈ 1.6280 . . . .
(3.50)
Generally K ∗
i > K ∗ . Thus, δ is a scaling parameter that characterizes the rate of
approach of the parameter K to its critical value, K ∗ .
MacKay also determined the position, p i , of the ith dominant elliptic fixed point
when K = K ∗
i . This gives the scaling parameter, β, along the p-axis, where
β = − lim
i→∞
p i+1 − p i
p i − p i−1
= −3.0668882.
(3.51)
The scaling parameter, α, along the x-axis can be found by finding the position, x i ,
of the elliptic point (in the same M i -cycle) nearest the dominant symmetry line and
then taking the limit
α = − lim
i→∞
x i−1
x i
= −1.4148360.
(3.52)
Thus, for K = K ∗ , the scaling matrix, B i , appears to take the limiting value
B
∗
= lim
i→∞
B i =
β 0
0 α
.
(3.53)