Reversible diffeomorphisms and flows

Devaney, Robert L.

doi:10.1090/S0002-9947-1976-0402815-3

Reversible diffeomorphisms and flows
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Trans. Amer. Math. Soc. 218 (1976), 89-113 Request permission

Abstract:

We generalize the classical notion of reversibility of a mechanical system. The generic qualitative properties of symmetric orbits of such systems are studied using transversality theory. In particular, we prove analogues of the closed orbit, Liapounov, and homoclinic orbit theorems for R-reversible systems.

References

Ralph Abraham, Jerrold E. Marsden, Al Kelley, and A. N. Kolmogorov, Foundations of mechanics. A mathematical exposition of classical mechanics with an introduction to the qualitative theory of dynamical systems and applications to the three-body problem, W. A. Benjamin, Inc., New York-Amsterdam, 1967. With the assistance of Jerrold E. Marsden; Four appendices, one by the author, two by Al Kelley, the fourth, a translation of an article by A. N. Kolmogorov. MR 0220467
Ralph Abraham and Joel Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York-Amsterdam, 1967. An appendix by Al Kelley. MR 0240836

The restricted problem of three bodies

39

George D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199–300. MR 1501070, DOI 10.1090/S0002-9947-1917-1501070-3
Martin Braun, Particle motions in a magnetic field, J. Differential Equations 8 (1970), 294–332. MR 264906, DOI 10.1016/0022-0396(70)90009-4
René De Vogelaere, On the structure of symmetric periodic solutions of conservative systems, with applications, Contributions to the theory of nonlinear oscillations, Vol. IV, Annals of Mathematics Studies, no. 41, Princeton University Press, Princeton, N.J., 1958, pp. 53–84. MR 0101362
René De Vogelaere, Surface de section dans le problème de Störmer, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 40 (1954), 705–714 (French). MR 0064632
Carlos Graef Fernández, Periodic orbits of the primary cosmic radiation, Bol. Soc. Mat. Mexicana 1 (1944), no. 3, 1–31 (Spanish). MR 0010826
Kenneth R. Meyer and Julian Palmore, A generic phenomenon in conservative Hamiltonian systems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 185–189. MR 0268914
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
Richard S. Palais, Equivalence of nearby differentiable actions of a compact group, Bull. Amer. Math. Soc. 67 (1961), 362–364. MR 130321, DOI 10.1090/S0002-9904-1961-10617-4
M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214–227. MR 209602, DOI 10.1016/0022-0396(67)90026-5
R. Clark Robinson, A global approximation theorem for Hamiltonian systems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 233–243. MR 0268915
R. Clark Robinson, Generic properties of conservative systems, Amer. J. Math. 92 (1970), 562–603. MR 273640, DOI 10.2307/2373361
Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Die Grundlehren der mathematischen Wissenschaften, Band 187, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme. MR 0502448, DOI 10.1007/978-3-642-87284-6
Ivan Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457–484 (French). MR 165536
S. Smale, Topology and mechanics. II. The planar $n$-body problem, Invent. Math. 11 (1970), 45–64. MR 321138, DOI 10.1007/BF01389805

Sur les trajectoires des corpuscules électrisés dans l’espace sous l’action du magnétisme terrestre avec application aux aurores boréales

24

Jacques Roels, An extension to resonant cases of Liapunov’s theorem concerning the periodic solutions near a Hamiltonian equilibrium, J. Differential Equations 9 (1971), 300–324. MR 273155, DOI 10.1016/0022-0396(71)90084-2

An outline of the allowed cone of cosmic radiation

Jacques Henrard, Proof of a conjecture of E. Strömgren, Celestial Mech. 7 (1973), 449–457. MR 365628, DOI 10.1007/BF01227510
J. Palis, On Morse-Smale dynamical systems, Topology 8 (1968), 385–404. MR 246316, DOI 10.1016/0040-9383(69)90024-X