60
3 Area-Preserving Maps
Fig. 3.9 Symmetry lines for
the standard map. The
symmetry line a is the
dominant symmetry line
because every elliptic
M-cycle will have a point
on it
p
x
and I 2
p
x
=
p
x
. For I 1 , the lines of fixed points are x = 0 and x =
1
2 ,
while, for I 2 , x =
p
2 and x =
p+1
2 are lines of fixed points. These lines are shown in
Fig. 3.9 and are denoted a, b, c, and d, respectively. At least two of the M points on
an M-cycle will be fixed points of I 1 or I 2 . There are two M-cycles for each winding
number ω =
N
M , one elliptic and one hyperbolic. Each M-cycle has M points. Of
the 2M fixed points with winding number ω =
N
M , four will lie on the lines a, b,
c, and d, with one on each line. Elliptic M-cycles will always have a point on the a
line, and therefore the a line is called the dominant symmetry line.
In Figs. 3.10, 3.11, and 3.12, we show a sequence of standard map plots for
increasing values of K. In Fig. 3.10a, we show a number of orbits of the standard
map for K = 0.1716354. The first thing to note is that the standard map has several
symmetry lines. Also, it repeats itself along both the p and x directions (mod 1), so
that all relevant information about the map is contained in the interval 0 ≤ p ≤ 0.5,
0 ≤ x ≤ 1.0. However, in the literature, various authors focus on different regions
of the standard map in the interval 0 ≤ p ≤ 1.0, so we show the entire interval
here. For K = 0.1716354, most of the map is composed of KAM tori. However, the
dominant resonances and periodic orbits are clearly seen. None of the separatrices
appear to be chaotic to the scale shown here.
In Fig. 3.10b, we show the standard map for K = 0.4716354. Again, the largest
resonances are clearly seen. The positions of the elliptic periodic orbits with winding
numbers ω =
0
1 and ω =
1
2 are unchanged as we increase K, but the other periodic
orbits are moved by the growing resonance zone surrounding the fixed point with
winding number ω =
0
1 . We note also that the separatrix for the hyperbolic orbit
with winding number ω =
0
1 begins to be chaotic in the neighborhood of the
hyperbolic fixed point. This plot is still dominated, however, by KAM tori.
In Fig. 3.11c, we show the structure of the standard map for K = 0.7716354.
Now the hyperbolic fixed points with winding numbers ω =
0
1 and ω =
1
2 appear
to have chaotic separatrices. All the resonances are becoming increasingly distorted
by the large resonance surrounding the ω =
0
1 elliptic point. There still are many
KAM tori stretching horizontally from x = 0 to x = 1.0 in this map.
3 Area-Preserving Maps
Fig. 3.9 Symmetry lines for
the standard map. The
symmetry line a is the
dominant symmetry line
because every elliptic
M-cycle will have a point
on it
p
x
and I 2
p
x
=
p
x
. For I 1 , the lines of fixed points are x = 0 and x =
1
2 ,
while, for I 2 , x =
p
2 and x =
p+1
2 are lines of fixed points. These lines are shown in
Fig. 3.9 and are denoted a, b, c, and d, respectively. At least two of the M points on
an M-cycle will be fixed points of I 1 or I 2 . There are two M-cycles for each winding
number ω =
N
M , one elliptic and one hyperbolic. Each M-cycle has M points. Of
the 2M fixed points with winding number ω =
N
M , four will lie on the lines a, b,
c, and d, with one on each line. Elliptic M-cycles will always have a point on the a
line, and therefore the a line is called the dominant symmetry line.
In Figs. 3.10, 3.11, and 3.12, we show a sequence of standard map plots for
increasing values of K. In Fig. 3.10a, we show a number of orbits of the standard
map for K = 0.1716354. The first thing to note is that the standard map has several
symmetry lines. Also, it repeats itself along both the p and x directions (mod 1), so
that all relevant information about the map is contained in the interval 0 ≤ p ≤ 0.5,
0 ≤ x ≤ 1.0. However, in the literature, various authors focus on different regions
of the standard map in the interval 0 ≤ p ≤ 1.0, so we show the entire interval
here. For K = 0.1716354, most of the map is composed of KAM tori. However, the
dominant resonances and periodic orbits are clearly seen. None of the separatrices
appear to be chaotic to the scale shown here.
In Fig. 3.10b, we show the standard map for K = 0.4716354. Again, the largest
resonances are clearly seen. The positions of the elliptic periodic orbits with winding
numbers ω =
0
1 and ω =
1
2 are unchanged as we increase K, but the other periodic
orbits are moved by the growing resonance zone surrounding the fixed point with
winding number ω =
0
1 . We note also that the separatrix for the hyperbolic orbit
with winding number ω =
0
1 begins to be chaotic in the neighborhood of the
hyperbolic fixed point. This plot is still dominated, however, by KAM tori.
In Fig. 3.11c, we show the structure of the standard map for K = 0.7716354.
Now the hyperbolic fixed points with winding numbers ω =
0
1 and ω =
1
2 appear
to have chaotic separatrices. All the resonances are becoming increasingly distorted
by the large resonance surrounding the ω =
0
1 elliptic point. There still are many
KAM tori stretching horizontally from x = 0 to x = 1.0 in this map.