Chapter 3
Area-Preserving Maps
Abstract Area-preserving maps can provide a picture of mechanisms causing the
transition to chaos. In non-integrable area-preserving twist maps, the separatrix
region of a nonlinear resonance contains a stochastic (chaotic) layer. Nonlinear
resonances are separated by KAM tori that become cantori as the transition to
global chaos occurs. The standard map is a twist map that describes the behavior
of separatrix regions, and it shows the universal scaling behavior and self-similarity
associated with the transition to chaos. This self-similarity is demonstrated explicitly by another twist map, called the universal map.
In systems for which it is not possible to construct an area-preserving map
analytically, the global behavior of the dynamics can be determined by working
directly with the Hamiltonian and using it to locate nonlinear resonance structures.
The Hamiltonian is then mapped between different spatial scales in the phase space
and a renormalization map is constructed for the amplitudes and relative wave
numbers of the resonances from one scale to the next. The renormalization map
allows one to determine parameter ranges for which chaos has developed in local
regions of the phase space.
Keywords Area preserving maps · Twist map · Standard map · Nonlinear
resonance · KAM tori · Universal map · Cantorus · Bifurcations ·
Self-similarity · Scaling · Renormalization map
3.1 Introduction
Area-preserving maps provide the simplest and most accurate means to visualize
and quantify the dynamical behavior of conservative systems with two degrees of
freedom. Such maps can be iterated on even the smallest computers with great
accuracy, and provide beautiful pictures of the mechanisms at play during the
transition to chaos.
The class of area-preserving maps we study in this chapter are twist maps.
When an integrable twist map is rendered nonintegrable by a small perturbation,
resonances can occur and degenerate lines of fixed points in the integrable map
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_3
47
Area-Preserving Maps
Abstract Area-preserving maps can provide a picture of mechanisms causing the
transition to chaos. In non-integrable area-preserving twist maps, the separatrix
region of a nonlinear resonance contains a stochastic (chaotic) layer. Nonlinear
resonances are separated by KAM tori that become cantori as the transition to
global chaos occurs. The standard map is a twist map that describes the behavior
of separatrix regions, and it shows the universal scaling behavior and self-similarity
associated with the transition to chaos. This self-similarity is demonstrated explicitly by another twist map, called the universal map.
In systems for which it is not possible to construct an area-preserving map
analytically, the global behavior of the dynamics can be determined by working
directly with the Hamiltonian and using it to locate nonlinear resonance structures.
The Hamiltonian is then mapped between different spatial scales in the phase space
and a renormalization map is constructed for the amplitudes and relative wave
numbers of the resonances from one scale to the next. The renormalization map
allows one to determine parameter ranges for which chaos has developed in local
regions of the phase space.
Keywords Area preserving maps · Twist map · Standard map · Nonlinear
resonance · KAM tori · Universal map · Cantorus · Bifurcations ·
Self-similarity · Scaling · Renormalization map
3.1 Introduction
Area-preserving maps provide the simplest and most accurate means to visualize
and quantify the dynamical behavior of conservative systems with two degrees of
freedom. Such maps can be iterated on even the smallest computers with great
accuracy, and provide beautiful pictures of the mechanisms at play during the
transition to chaos.
The class of area-preserving maps we study in this chapter are twist maps.
When an integrable twist map is rendered nonintegrable by a small perturbation,
resonances can occur and degenerate lines of fixed points in the integrable map
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_3
47