72
3 Area-Preserving Maps
From the above, we conclude that a periodic orbit, x 0 , can only be created or
destroyed or can collide with another periodic orbit of the same period when the
tangent map, ∇T K (x), has an eigenvalue λ = 1. We can generalize this statement to
include fixed points of the mapping, T M
K (x), for M integer. Fixed points of T M
K (x)
are isolated unless the tangent map, ∇T M
K (x), has an eigenvalue λ = 1 or unless
∇T K (x) has an eigenvalue λ such that λ M = 1. If λ M = 1, then fixed points of
T M
K (x) can be created or destroyed or can collide with another fixed point. The case
M = 2 corresponds to period-doubling bifurcations. Bifurcations also occur for
M > 2 but are not as important as the period-doubling bifurcations because perioddoubling bifurcations are the last to occur before a stable periodic orbit completely
loses stability and renders the region of phase space in its neighborhood totally
chaotic (MacKay 1983b). We shall only consider period-doubling bifurcations here,
but discussion of other types that can occur may be found in Meyer (1970); Collet
et al. (1981); Greene et al. (1981).
3.6.2 The Quadratic Map
It is interesting to illustrate some of these ideas for the quadratic map, Q a , defined as
y n+1
x n+1
= Q a
y n
x n
=
x n
1 − y n − ax 2
n
,
(3.55)
where a is the parameter of the map. Quadratic maps are especially important
because they approximate the behavior of the neighborhood of stable islands in the
chaotic sea of more general conservative maps.
Quadratic maps generally consist of one stable island surrounded by a chaotic
sea. The quadratic map in Eq. (3.55) is reversible and can be written as the product
of two involutions, Q a = S 2 S 1 , where S 2
1 = S 2
2 = I and I is the identity map. The
involutions S 1 and S 2 are defined as
y n+1
x n+1
= S 1
y n
x n
=
x n
y n
(3.56)
and
y n+1
x n+1
= S 2
y n
x n
=
y n
1 − x n − ay 2
n
.
(3.57)
The symmetry lines for this map are the lines of fixed points of the maps S 1 and S 2
and are given by x = y and 2x − 1 + ay 2 = 0, respectively. Thus, the orbits of the
quadratic map are symmetric about an axis making an angle of 45 o with the x-axis
(and the y-axis).
3 Area-Preserving Maps
From the above, we conclude that a periodic orbit, x 0 , can only be created or
destroyed or can collide with another periodic orbit of the same period when the
tangent map, ∇T K (x), has an eigenvalue λ = 1. We can generalize this statement to
include fixed points of the mapping, T M
K (x), for M integer. Fixed points of T M
K (x)
are isolated unless the tangent map, ∇T M
K (x), has an eigenvalue λ = 1 or unless
∇T K (x) has an eigenvalue λ such that λ M = 1. If λ M = 1, then fixed points of
T M
K (x) can be created or destroyed or can collide with another fixed point. The case
M = 2 corresponds to period-doubling bifurcations. Bifurcations also occur for
M > 2 but are not as important as the period-doubling bifurcations because perioddoubling bifurcations are the last to occur before a stable periodic orbit completely
loses stability and renders the region of phase space in its neighborhood totally
chaotic (MacKay 1983b). We shall only consider period-doubling bifurcations here,
but discussion of other types that can occur may be found in Meyer (1970); Collet
et al. (1981); Greene et al. (1981).
3.6.2 The Quadratic Map
It is interesting to illustrate some of these ideas for the quadratic map, Q a , defined as
y n+1
x n+1
= Q a
y n
x n
=
x n
1 − y n − ax 2
n
,
(3.55)
where a is the parameter of the map. Quadratic maps are especially important
because they approximate the behavior of the neighborhood of stable islands in the
chaotic sea of more general conservative maps.
Quadratic maps generally consist of one stable island surrounded by a chaotic
sea. The quadratic map in Eq. (3.55) is reversible and can be written as the product
of two involutions, Q a = S 2 S 1 , where S 2
1 = S 2
2 = I and I is the identity map. The
involutions S 1 and S 2 are defined as
y n+1
x n+1
= S 1
y n
x n
=
x n
y n
(3.56)
and
y n+1
x n+1
= S 2
y n
x n
=
y n
1 − x n − ay 2
n
.
(3.57)
The symmetry lines for this map are the lines of fixed points of the maps S 1 and S 2
and are given by x = y and 2x − 1 + ay 2 = 0, respectively. Thus, the orbits of the
quadratic map are symmetric about an axis making an angle of 45 o with the x-axis
(and the y-axis).