3.1 Introduction
49
As we vary the parameters of a conservative map, an elliptic fixed point may
bifurcate and change to a hyperbolic fixed point (this is called tangent bifurcation)
or may bifurcate into several new fixed points. The case when it bifurcates into
one hyperbolic fixed point and two elliptic fixed points is called a period-doubling
bifurcation. As the mapping parameter is varied, sequences of period-doubling
bifurcations may occur. Such bifurcation sequences also exhibit universal scaling
behavior. In Sect. 3.6, we will give criteria to determine at what mapping parameter
values such bifurcations can occur.
In Sect. 3.7, we will focus on cantori, which are the remnants of KAM tori just
after they are destroyed. The mechanism by which phase space trajectories pass
through cantori resembles that of turnstiles or revolving doors that allow a two-way
flow of traffic. Turnstiles can be associated with the rational approximates to the
KAM torus that has been destroyed. They pump an amount of phase space area that
itself exhibits scaling behavior.
In practice, we are often confronted with a physical system whose Hamiltonian
we are given, and then we must determine as much as possible about its global
dynamics. We have to ask: What regions of the phase space might undergo a
transition to chaos and for what parameter values does it happen? For such systems,
it is usually not possible to construct an area-preserving map analytically, but there
is still a great deal we can learn about the global behavior by working directly with
the Hamiltonian.
The global properties of a Hamiltonian system are determined by the location
and size of its nonlinear resonances. However, in systems with time-independent
Hamiltonians, it is not always easy to locate the nonlinear resonances. There is some
hope if the Hamiltonian separates into an integrable part, for which action-angle
variables can be found, and a perturbation that renders the total Hamiltonian nonintegrable.
Escande and Doveil (1981); Escande (1982, 1985) developed a renormalization
scheme, based on a mapping of Hamiltonians between different spatial scales in
the phase space, to describe the destruction of KAM tori. The starting point of this
theory is a Hamiltonian that allows one to identify the primary nonlinear resonances
of the system. We can then focus on a KAM torus that will be most strongly affected
by two primary resonances that bracket it. We then approximate the full Hamiltonian
by a Hamiltonian containing only those two resonances. We then repeat the process
and thereby generate a mapping of Hamiltonians for the pairs of resonances that
bracket the same KAM torus on each scale. It is possible to write a renormalization
map for the amplitudes and relative wave numbers of the resonances from one scale
to the next. This renormalization map allows us to determine if the KAM torus
is destroyed or not for the given parameters of the system. In Sect. 3.8, we derive
the renormalization map. Finally, in Sect. 3.8.2, we study the fixed points of the
renormalization map.
49
As we vary the parameters of a conservative map, an elliptic fixed point may
bifurcate and change to a hyperbolic fixed point (this is called tangent bifurcation)
or may bifurcate into several new fixed points. The case when it bifurcates into
one hyperbolic fixed point and two elliptic fixed points is called a period-doubling
bifurcation. As the mapping parameter is varied, sequences of period-doubling
bifurcations may occur. Such bifurcation sequences also exhibit universal scaling
behavior. In Sect. 3.6, we will give criteria to determine at what mapping parameter
values such bifurcations can occur.
In Sect. 3.7, we will focus on cantori, which are the remnants of KAM tori just
after they are destroyed. The mechanism by which phase space trajectories pass
through cantori resembles that of turnstiles or revolving doors that allow a two-way
flow of traffic. Turnstiles can be associated with the rational approximates to the
KAM torus that has been destroyed. They pump an amount of phase space area that
itself exhibits scaling behavior.
In practice, we are often confronted with a physical system whose Hamiltonian
we are given, and then we must determine as much as possible about its global
dynamics. We have to ask: What regions of the phase space might undergo a
transition to chaos and for what parameter values does it happen? For such systems,
it is usually not possible to construct an area-preserving map analytically, but there
is still a great deal we can learn about the global behavior by working directly with
the Hamiltonian.
The global properties of a Hamiltonian system are determined by the location
and size of its nonlinear resonances. However, in systems with time-independent
Hamiltonians, it is not always easy to locate the nonlinear resonances. There is some
hope if the Hamiltonian separates into an integrable part, for which action-angle
variables can be found, and a perturbation that renders the total Hamiltonian nonintegrable.
Escande and Doveil (1981); Escande (1982, 1985) developed a renormalization
scheme, based on a mapping of Hamiltonians between different spatial scales in
the phase space, to describe the destruction of KAM tori. The starting point of this
theory is a Hamiltonian that allows one to identify the primary nonlinear resonances
of the system. We can then focus on a KAM torus that will be most strongly affected
by two primary resonances that bracket it. We then approximate the full Hamiltonian
by a Hamiltonian containing only those two resonances. We then repeat the process
and thereby generate a mapping of Hamiltonians for the pairs of resonances that
bracket the same KAM torus on each scale. It is possible to write a renormalization
map for the amplitudes and relative wave numbers of the resonances from one scale
to the next. This renormalization map allows us to determine if the KAM torus
is destroyed or not for the given parameters of the system. In Sect. 3.8, we derive
the renormalization map. Finally, in Sect. 3.8.2, we study the fixed points of the
renormalization map.