66
3 Area-Preserving Maps
3.5.1 Integrable Twist Map
The rational approximates to the inverse golden mean KAM torus all have one
elliptic periodic orbit on the dominant symmetry line (see Fig. 3.9).
Pick one particular elliptic periodic orbit P M
0 of an M i -cycle with winding
number ω i =
N i
M i
on the dominant symmetry line. Construct a map for which that
periodic orbit is a fixed point. The point P M
0 can be written
P
M
0 =
p 0
x 0
=
ω i + v i (t 0 )
t 0 + u i (t 0 )
.
(3.31)
where v i and u i are periodic with period one and ω i =
N i
M i
.
Let G denote a twist map that has an inverse golden mean KAM torus. (The
standard map is just one example of many such maps.) Then, under the mapping
G M i , the point P M
0 becomes
G
M i
p 0
x 0
=
ω i + v i (t 0 + M i ω i )
t 0 + M i ω i + u i (t 0 + M i ω i )
.
(3.32)
In order to return to our original point, we must introduce the mapping R, defined as
R
p 0
x 0
=
p 0
x 0 − 1
,
(3.33)
which commutes with G. Then let the map R act N i times to get
G
M i R
N i
p 0
x 0
=
ω i + v i (t 0 + M i ω i )
t 0 + M i ω i − N i + u i (t 0 + M i ω i )
=
ω i + v i (t 0 )
t 0 + u i (t 0 )
(3.34)
since v i and u i are periodic with period one and ω i =
N i
M i
. Thus, the mapping
G M i R N i maps a point on the M i -cycle onto itself and maps a neighborhood of that
point back to itself.
Are the neighborhoods of the elliptic fixed points on the dominant symmetry line
(the dominant elliptic fixed points) self-similar? If we rescale each neighborhood,
do we get exactly the same picture back?
In order to determine how to rescale the neighborhoods of the dominant elliptic
fixed points, begin the process with an integrable twist map. Consider the standard
map for K = 0, which can be written p n+1 = p n and x n+1 = x n + p n+1 , with
boundary condition, 0≤x≤1 mod(1). The M i -cycle with winding number ω i =
N i
M i
=
F i−1
F i
has a dominant elliptic fixed point with coordinates p i = ω i and x i =
0. The distance along the p-axis, p i , between the fixed points at p = ω i and
p = ω i+1 is p i = ω i+1 − ω i . The distance along the x-axis between the dominant
elliptic fixed point at x = 0 and its closest neighboring elliptic fixed point belonging
3 Area-Preserving Maps
3.5.1 Integrable Twist Map
The rational approximates to the inverse golden mean KAM torus all have one
elliptic periodic orbit on the dominant symmetry line (see Fig. 3.9).
Pick one particular elliptic periodic orbit P M
0 of an M i -cycle with winding
number ω i =
N i
M i
on the dominant symmetry line. Construct a map for which that
periodic orbit is a fixed point. The point P M
0 can be written
P
M
0 =
p 0
x 0
=
ω i + v i (t 0 )
t 0 + u i (t 0 )
.
(3.31)
where v i and u i are periodic with period one and ω i =
N i
M i
.
Let G denote a twist map that has an inverse golden mean KAM torus. (The
standard map is just one example of many such maps.) Then, under the mapping
G M i , the point P M
0 becomes
G
M i
p 0
x 0
=
ω i + v i (t 0 + M i ω i )
t 0 + M i ω i + u i (t 0 + M i ω i )
.
(3.32)
In order to return to our original point, we must introduce the mapping R, defined as
R
p 0
x 0
=
p 0
x 0 − 1
,
(3.33)
which commutes with G. Then let the map R act N i times to get
G
M i R
N i
p 0
x 0
=
ω i + v i (t 0 + M i ω i )
t 0 + M i ω i − N i + u i (t 0 + M i ω i )
=
ω i + v i (t 0 )
t 0 + u i (t 0 )
(3.34)
since v i and u i are periodic with period one and ω i =
N i
M i
. Thus, the mapping
G M i R N i maps a point on the M i -cycle onto itself and maps a neighborhood of that
point back to itself.
Are the neighborhoods of the elliptic fixed points on the dominant symmetry line
(the dominant elliptic fixed points) self-similar? If we rescale each neighborhood,
do we get exactly the same picture back?
In order to determine how to rescale the neighborhoods of the dominant elliptic
fixed points, begin the process with an integrable twist map. Consider the standard
map for K = 0, which can be written p n+1 = p n and x n+1 = x n + p n+1 , with
boundary condition, 0≤x≤1 mod(1). The M i -cycle with winding number ω i =
N i
M i
=
F i−1
F i
has a dominant elliptic fixed point with coordinates p i = ω i and x i =
0. The distance along the p-axis, p i , between the fixed points at p = ω i and
p = ω i+1 is p i = ω i+1 − ω i . The distance along the x-axis between the dominant
elliptic fixed point at x = 0 and its closest neighboring elliptic fixed point belonging