50
3 Area-Preserving Maps
3.2 Twist Maps
Twist maps are area-preserving maps that provide a clear visualization of dynamical
systems with two degrees of freedom. They provide an analytic representation
of a Poincaré surface of section. Twist maps may be integrable or nonintegrable.
Birkhoff’s fixed point theorem describes changes that occur in an integrable map
when its integrability is destroyed by a perturbation. These changes are described
in general terms in this section and in more detail throughout the remainder of this
chapter.
3.2.1 Derivation of a Twist Map from a Torus
A twist map is an area-preserving map that can be derived from a Poincaré surface
of section of the torus. To see how this is done, let us consider the torus shown in
Fig. 3.1. Each time the trajectory passes a given angle θ 2 = we plot its position in
the (J 1 , θ 1 ) plane. For an integrable system, the plotted points will lie on a circle of
radius
√
2J 1 , and the time interval between passes will be τ =
2π
ω 2
, where ˙
θ 2 = ω 2 =
ω 2 (J 1 , J 2 ). Let us now assume that the nth passage of the trajectory occurs at angle
θ 1 = φ n . Then the (n + 1)st passage will occur at angle θ 1 = φ n+1 = φ n + ω 1 τ =
θ 0 + 2π
ω 1
ω 2
. The quantity ω =
ω 1
ω 2
is called the winding number. The area enclosed
by the circle is 2πJ 1 . Now introduce the coordinate, ρ, where 2πJ 1 = πρ 2 , and ρ is
the radius of the circle, ρ =
√
2J 1 . For a given energy E, the value of ρ determines
J 2 as well. Our mapping now takes the form
ρ n+1 = ρ n ,
φ n+1 = φ n + 2πω(ρ n ),
(3.1)
where ω is assumed to be a smooth function of ρ n .
Fig. 3.1 For integrable systems, the twist map consists of trajectories that densely fill a circle
(irrational winding number ω) and discrete, periodic points (rational winding number ω). The rate
at which a trajectory completes one revolution of the circle depends on the radius. Thus an initial
line of points, a, becomes twisted, b, by the map
3 Area-Preserving Maps
3.2 Twist Maps
Twist maps are area-preserving maps that provide a clear visualization of dynamical
systems with two degrees of freedom. They provide an analytic representation
of a Poincaré surface of section. Twist maps may be integrable or nonintegrable.
Birkhoff’s fixed point theorem describes changes that occur in an integrable map
when its integrability is destroyed by a perturbation. These changes are described
in general terms in this section and in more detail throughout the remainder of this
chapter.
3.2.1 Derivation of a Twist Map from a Torus
A twist map is an area-preserving map that can be derived from a Poincaré surface
of section of the torus. To see how this is done, let us consider the torus shown in
Fig. 3.1. Each time the trajectory passes a given angle θ 2 = we plot its position in
the (J 1 , θ 1 ) plane. For an integrable system, the plotted points will lie on a circle of
radius
√
2J 1 , and the time interval between passes will be τ =
2π
ω 2
, where ˙
θ 2 = ω 2 =
ω 2 (J 1 , J 2 ). Let us now assume that the nth passage of the trajectory occurs at angle
θ 1 = φ n . Then the (n + 1)st passage will occur at angle θ 1 = φ n+1 = φ n + ω 1 τ =
θ 0 + 2π
ω 1
ω 2
. The quantity ω =
ω 1
ω 2
is called the winding number. The area enclosed
by the circle is 2πJ 1 . Now introduce the coordinate, ρ, where 2πJ 1 = πρ 2 , and ρ is
the radius of the circle, ρ =
√
2J 1 . For a given energy E, the value of ρ determines
J 2 as well. Our mapping now takes the form
ρ n+1 = ρ n ,
φ n+1 = φ n + 2πω(ρ n ),
(3.1)
where ω is assumed to be a smooth function of ρ n .
Fig. 3.1 For integrable systems, the twist map consists of trajectories that densely fill a circle
(irrational winding number ω) and discrete, periodic points (rational winding number ω). The rate
at which a trajectory completes one revolution of the circle depends on the radius. Thus an initial
line of points, a, becomes twisted, b, by the map