48
3 Area-Preserving Maps
are changed to finite chains of alternating hyperbolic and elliptic fixed points
surrounded by nonlinear resonance zones. As the strength of the perturbation is
increased, the resonance zones can grow and overlap and form a chaotic sea.
Chaos appears first in the neighborhood of hyperbolic fixed points and is due
to an incredibly complex dynamics that occurs in that neighborhood. In fact, near
hyperbolic fixed points it is possible to embed a Bernoulli shift with an infinite
alphabet in local regions of the phase space. This means that these regions are Kflows and therefore are chaotic.
For small mapping parameters, resonance zones are separated from one another
by KAM tori. In area-preserving maps with two degrees of freedom, KAM tori
serve to isolate one region of the phase space from another. KAM tori can be
destroyed by nonlinear resonances. The mechanism by which this occurs is quite
beautiful. Each KAM torus has an irrational winding number (winding numbers are
defined in Sect. 3.2). Resonance zones form island chains, and each island chain
contains a sequence of hyperbolic and elliptic fixed points that have a rationalfraction winding number. Greene (1979a) has shown that each KAM torus can
be approximated by a unique sequence of island chains whose winding numbers
are given by the continued fraction representing the irrational winding number of
the KAM torus. The island chains that approximate a given KAM torus play a
dominant role in its destruction. A KAM torus is destroyed suddenly, as the mapping
parameter increases, and forms a cantorus. A cantorus can still partially block the
flow of trajectories in phase space. As the mapping parameter is increased further,
the cantorus gradually disappears and trajectories are free to diffuse more or less at
random in the chaotic sea.
In the subsequent sections of this chapter, we will describe in detail the intricate
behavior associated with the transition to chaos in area-preserving maps. We will
begin in Sect. 3.2 by describing the general behavior of twist maps. Twist maps
characterize well the behavior of conservative systems with two degrees of freedom.
We will use them throughout this book.
In nonintegrable twist maps, the separatrix region associated with nonlinear
resonances contains a stochastic (chaotic) layer. In Sect. 3.3, we derive a map, the
standard map, which is a map that describes the behavior of the separatrix region.
We will use the standard map to show the mechanism by which KAM tori are
destroyed, and we demonstrate the scaling behavior associated with the destruction
of KAM tori. In 3.4, we then focus on the scaling behavior of “noble” KAM tori,
which are the last KAM tori to be destroyed in a global transition to chaos.
The detailed mechanism by which a KAM torus is destroyed is determined by its
winding number and not the particular twist map that it belongs to. For this reason,
the destruction of a KAM torus and its change into a cantorus is associated with
universal scaling behavior. The phase space associated with this process shows selfsimilarity. MacKay (1982, 1983a) has constructed a map, called the universal map,
that shows explicitly this self-similar behavior. In Sect. 3.5, we describe properties
of the universal map.
3 Area-Preserving Maps
are changed to finite chains of alternating hyperbolic and elliptic fixed points
surrounded by nonlinear resonance zones. As the strength of the perturbation is
increased, the resonance zones can grow and overlap and form a chaotic sea.
Chaos appears first in the neighborhood of hyperbolic fixed points and is due
to an incredibly complex dynamics that occurs in that neighborhood. In fact, near
hyperbolic fixed points it is possible to embed a Bernoulli shift with an infinite
alphabet in local regions of the phase space. This means that these regions are Kflows and therefore are chaotic.
For small mapping parameters, resonance zones are separated from one another
by KAM tori. In area-preserving maps with two degrees of freedom, KAM tori
serve to isolate one region of the phase space from another. KAM tori can be
destroyed by nonlinear resonances. The mechanism by which this occurs is quite
beautiful. Each KAM torus has an irrational winding number (winding numbers are
defined in Sect. 3.2). Resonance zones form island chains, and each island chain
contains a sequence of hyperbolic and elliptic fixed points that have a rationalfraction winding number. Greene (1979a) has shown that each KAM torus can
be approximated by a unique sequence of island chains whose winding numbers
are given by the continued fraction representing the irrational winding number of
the KAM torus. The island chains that approximate a given KAM torus play a
dominant role in its destruction. A KAM torus is destroyed suddenly, as the mapping
parameter increases, and forms a cantorus. A cantorus can still partially block the
flow of trajectories in phase space. As the mapping parameter is increased further,
the cantorus gradually disappears and trajectories are free to diffuse more or less at
random in the chaotic sea.
In the subsequent sections of this chapter, we will describe in detail the intricate
behavior associated with the transition to chaos in area-preserving maps. We will
begin in Sect. 3.2 by describing the general behavior of twist maps. Twist maps
characterize well the behavior of conservative systems with two degrees of freedom.
We will use them throughout this book.
In nonintegrable twist maps, the separatrix region associated with nonlinear
resonances contains a stochastic (chaotic) layer. In Sect. 3.3, we derive a map, the
standard map, which is a map that describes the behavior of the separatrix region.
We will use the standard map to show the mechanism by which KAM tori are
destroyed, and we demonstrate the scaling behavior associated with the destruction
of KAM tori. In 3.4, we then focus on the scaling behavior of “noble” KAM tori,
which are the last KAM tori to be destroyed in a global transition to chaos.
The detailed mechanism by which a KAM torus is destroyed is determined by its
winding number and not the particular twist map that it belongs to. For this reason,
the destruction of a KAM torus and its change into a cantorus is associated with
universal scaling behavior. The phase space associated with this process shows selfsimilarity. MacKay (1982, 1983a) has constructed a map, called the universal map,
that shows explicitly this self-similar behavior. In Sect. 3.5, we describe properties
of the universal map.