Numerical study of quadratic area-preserving mappings
Author:
M. Hénon
Journal:
Quart. Appl. Math. 27 (1969), 291-312
MSC:
Primary 65.10
DOI:
https://doi.org/10.1090/qam/253513
MathSciNet review:
253513
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Abstract: Dynamical systems with two degrees of freedom can be reduced to the study of an area-preserving mapping. We consider here, as a model problem, the mapping given by the quadratic equations: ${x_1} = x\cos \alpha - \left ( {y - {x^2}} \right )\sin \alpha$, ${y_1} = x\sin \alpha + \left ( {y - {x^2}} \right ) \\ \cos \alpha$, which is shown to be in a sense the simplest nontrivial mapping. Some analytical properties are given, and numerical results are exhibited in Figs. 2 to 14.
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M. Henon, Exploration numérique du problème restreint. III, IV, Bull. Astron. (série 3) 1, fasc. 1, 57–79 and fasc. 2, 49–66 (1966)
M. Henon and C. Heiles, The applicability of the third integral of motion: some numerical experiments, Astronom. J. 69, 73–79 (1964)
W. T. Kyner, Qualitative properties of orbits around an oblate planet, Comm. Pure Appl. Math. 17, 227–236 (1964)
V. A. Zlatoustov, D. E. Okhotsimskii, V. A. Sarychev, A. P. Torzhevskii, Oscillations of a satellite in the plane of an elliptic orbit, Kosmicheskie Issledovaniya 2, 657–666 (1964) (Russian)
A. J. Dragt, Trapped orbits in a magnetic dipole field, Rev. Geophys. 3, 255–298 (1965)
A. Garren, R. J. Riddell, L. Smith, G. Bing, L. R. Henrich, T. G. Northrop, J. E. Roberts, Individual particle motion and the effects of scattering in an axially symmetric magnetic field, Proc. Second U. N. International Conf. on Peaceful Uses of Atomic Energy (United Nations, Geneva) 31, 65–71 (1958)
T. G. Northrop, The adiabatic motion of charged particles, Interscience Publishers, New York, 1963, Chap. 5
G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18, 199–300 (1917)
G. D. Birkhoff, Surface transformations and their dynamical applications, Acta Math. 43,1–119 (1920)
J. Roels and M. Henon, Recherche des courbes invariantes d’une transformation ponctuelle plane conservant les aires, Bull. Astronom. (série 3) 2, 267–285 (1967)
C. L. Siegel, Vorlesungen über Himmelsmechanik, Springer-Verlag, Berlin, 1956, p. 203
V. I. Arnol’d, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Uspehi mat. Nauk 18, no. 5, 13–40 = Russian Math. Surveys 18, no. 5, 9–36 (1963)
J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., no. 1, 1–20 (1962)
V. I. Arnold and A. Avez, Problèmes ergodiques de la mécanique classique, Gauthier-Villars, Paris, 1967
G. Contopoulos, Resonance phenomena and the nonapplicability of the “third” integral, Bull. Astronom. (série 3) 2, 223–241 (1967)
B. V. Chirikov, When does the dynamical system turn into the statistical one?, Comm. at the International Congress of Mathematicians, Moscow, 1966
J. Moser, Stability of the asteroids, Astronom. J. 63, 439–443 (1958)
W. Engel, Ein Satz über ganze Cremona-Transformationen der Ebene, Math. Ann. 130, 11–19 (1955)
W. Engel, Ganze Cremona-Transformationen von Primzahlgrad in der Ebene, Math. Ann. 136, 319–325 (1958)
J. Moser, On the integrability of area-preserving Cremona mappings near an elliptic fixed point, Bol. Soc. Mat. Mexicana, 176–180 (1960)
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Article copyright:
© Copyright 1969
American Mathematical Society
