96
3 Area-Preserving Maps
where P n is the inverse semimajor axis, and g n is the phase of the orbit of Jupiter,
when the comet passes the perihelion. K is a constant determined by the mass of
Jupiter and the strength of the coupling between Jupiter and the comet. When P >
0, P = 0, or P < 0, the comet orbit is hyperbolic, parabolic, or elliptic, respectively.
For the parameters relevant to this system, the map shows chaos at the border P = 0,
indicating that there is chaotic capture and expulsion of comets in the solar system.
In this chapter, we have focused primarily on the behavior of twist maps.
However, in hydrodynamic flows and plasmas, dynamics has been observed that can
be understood in terms of maps that violate the twist condition. Properties of these
so-called nontwist maps have been discussed by del-Castillo-Negrete and Morrison
(1993); del-Castillo-Negrete et al. (1996a,b).
As we have seen, when maps become fully chaotic, their trajectories begin to
behave very much like simple random walks. The relation of fully chaotic maps to
random walks has been shown rigorously for some fully chaotic maps. For these
maps (the baker’s map is one example) the spectral properties can be obtained
exactly and analytic expressions have been obtained for the evolution of probability
distributions and diffusion coefficients. A very clear discussion of the spectral
analysis of fully chaotic maps, and related references, can be found in Driebe (1999).
In this chapter, we have studied nonlinear conservative systems with two degrees
of freedom from the point of view of planar area-preserving maps. In Chap. 5 we
will also show that there is a qualitatively different topology of chaos in systems
with three or more degrees of freedom.
References
Aubry S (1978) Solitons and condensed matter physics. In: Bishop AR, Schneider T (ed). Springer,
Berlin, p 264
Aubry S, LeDaeron PY (1983) Physica D 8:381
Bensimon D, Kadanoff LP (1984) Physics D 13:82
Berry MV (1978) AIP conference proceedings, vol 46. American Institute of Physics, New York,
p 16. Reprinted in [MacKay and Meiss 1987]
Birkhoff GD (1927) Acta Math 50:359. Reprinted in [MacKay and Meiss 1987]
Bountis T (1981) Physica D 3:577
Casati G, Guarneri I, Shepelyansky DL (1988) IEEE J Quantum Electron 24:1420
Chirikov B, Shepelyansky DL (1986) Chaos border and statistical anomalies. Preprint 86–174.
Institute of Nuclear Physics, Novosibirsk
Cocke S, Reichl LE (1990) Phys Rev A 41:3733
Collet P, Eckmann J-P, Koch H (1981) Physica D 3:457
del-Castillo-Negrete D, Morrison PJ (1993) Phys. Fluids A 5:948
del-Castillo-Negrete D, Greene JM, Morrison PJ (1996a) Physica D 91:1
del-Castillo-Negrete D, Greene JM, Morrison PJ (1996b) Physica D 100:311
de Vogelaere R (1958) In: Lefschetz S (ed) Contributions to the theory of nonlinear oscillations,
vol. IV. Princeton University Press, Princeton, p 53
Driebe DJ (1999) Fully chaotic maps and broken time symmetry. Kluwer Academic Publishers,
Dordrecht
Escande DF (1982) Phys. Scr. T2/1:126
3 Area-Preserving Maps
where P n is the inverse semimajor axis, and g n is the phase of the orbit of Jupiter,
when the comet passes the perihelion. K is a constant determined by the mass of
Jupiter and the strength of the coupling between Jupiter and the comet. When P >
0, P = 0, or P < 0, the comet orbit is hyperbolic, parabolic, or elliptic, respectively.
For the parameters relevant to this system, the map shows chaos at the border P = 0,
indicating that there is chaotic capture and expulsion of comets in the solar system.
In this chapter, we have focused primarily on the behavior of twist maps.
However, in hydrodynamic flows and plasmas, dynamics has been observed that can
be understood in terms of maps that violate the twist condition. Properties of these
so-called nontwist maps have been discussed by del-Castillo-Negrete and Morrison
(1993); del-Castillo-Negrete et al. (1996a,b).
As we have seen, when maps become fully chaotic, their trajectories begin to
behave very much like simple random walks. The relation of fully chaotic maps to
random walks has been shown rigorously for some fully chaotic maps. For these
maps (the baker’s map is one example) the spectral properties can be obtained
exactly and analytic expressions have been obtained for the evolution of probability
distributions and diffusion coefficients. A very clear discussion of the spectral
analysis of fully chaotic maps, and related references, can be found in Driebe (1999).
In this chapter, we have studied nonlinear conservative systems with two degrees
of freedom from the point of view of planar area-preserving maps. In Chap. 5 we
will also show that there is a qualitatively different topology of chaos in systems
with three or more degrees of freedom.
References
Aubry S (1978) Solitons and condensed matter physics. In: Bishop AR, Schneider T (ed). Springer,
Berlin, p 264
Aubry S, LeDaeron PY (1983) Physica D 8:381
Bensimon D, Kadanoff LP (1984) Physics D 13:82
Berry MV (1978) AIP conference proceedings, vol 46. American Institute of Physics, New York,
p 16. Reprinted in [MacKay and Meiss 1987]
Birkhoff GD (1927) Acta Math 50:359. Reprinted in [MacKay and Meiss 1987]
Bountis T (1981) Physica D 3:577
Casati G, Guarneri I, Shepelyansky DL (1988) IEEE J Quantum Electron 24:1420
Chirikov B, Shepelyansky DL (1986) Chaos border and statistical anomalies. Preprint 86–174.
Institute of Nuclear Physics, Novosibirsk
Cocke S, Reichl LE (1990) Phys Rev A 41:3733
Collet P, Eckmann J-P, Koch H (1981) Physica D 3:457
del-Castillo-Negrete D, Morrison PJ (1993) Phys. Fluids A 5:948
del-Castillo-Negrete D, Greene JM, Morrison PJ (1996a) Physica D 91:1
del-Castillo-Negrete D, Greene JM, Morrison PJ (1996b) Physica D 100:311
de Vogelaere R (1958) In: Lefschetz S (ed) Contributions to the theory of nonlinear oscillations,
vol. IV. Princeton University Press, Princeton, p 53
Driebe DJ (1999) Fully chaotic maps and broken time symmetry. Kluwer Academic Publishers,
Dordrecht
Escande DF (1982) Phys. Scr. T2/1:126