Abstract
A general KAM-theory for reversible systems is given. The cases of both maximal and lower-dimensional tori are covered. In some cases parameters are needed for persistence, therefore an unfolding theory is developed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V. I. (1971). On matrices depending on parameters.Russ. Math. Surv. 26(2), 29–43.
Arnold, V. I. (1984). Reversible systems. InNonlinear and Turbulent Processes in Physics, Academic, New York, Vol. 3, pp. 1161–1174.
Arnold, V. I., and Sevryuk, M. B. (1986). Oscillations and bifurcations in reversible systems. InNonlinear Phenomena in Plasma Physics and Hydrodynamics, Mir, Moscow, pp. 31–64.
Bibikov, Yu. N. (1991).Multifrequency Nonlinear Oscillations and Their Bifurcations, Leningrad University Press, Leningrad (in Russian).
Braaksma, B. L. J., Broer, H. W., and Huitema, G. B. (1990). Toward a quasi-periodic bifurcation theory.Mem. Am. Math. Soc. 421, 83–175.
Broer, H. W., Chow, S.-N., Kim, Y., and Vegter, G. (1993). A normally elliptic Hamiltonian bifurcation.Z. Angew. Math. Phys. 44, 389–432.
Broer, H. W., Huitema, G. B., and Takens, F. (1990). Unfoldings of quasi-periodic tori.Mem. Am. Math. Soc. 421, 1–82.
Huitema, G. B. (1988).Unfoldings of Quasi-periodic Tori, Thesis, University of Groningen, Groningen.
Moser, J. K. (1966). On the theory of quasiperiodic motions.SIAM Rev. 8, II, 145–172.
Moser, J. K. (1967). Convergent series expansions for quasi-periodic motions.Math. Ann. 169, 136–176.
Moser, J. K. (1973).Stable and Random Motions in Dynamical Systems, Ann. Math. Stud. 77, Princeton University Press, Princeton, NJ.
Parasyuk, I. O. (1982). Preservation of quasiperiodic motions in reversible multifrequency systems.Dokl. Akad. Nauk Ukrain. SSR Ser. A 9, 19–22 (in Russian).
Pöschel, J. (1982). Integrability of Hamiltonian systems on Cantor sets.Comm. Pure Appl. Math. 35, 653–696.
Quispel, G. R. W., and Sevryuk, M. B. (1993). KAM theorems for the product of two involutions of different types.Chaos 3(4), 757–769.
Scheurle, J. (1987). Bifurcations of quasi-periodic solutions from equilibrium points of reversible dynamical systems.Arch. Rat. Mech. Anal. 97(2), 103–139.
Sevryuk, M. B. (1986).Reversible Systems, LNM 1211, Springer-Verlag, New York.
Sevryuk, M. B. (1990). Invariantm-dimensional tori of reversible systems with a phase space of dimension greater than 2m.J. Sov. Math. 51(3), 2374–2386.
Sevryuk, M. B. (1991a). Lower dimensional tori in reversible systems.Chaos 1(2), 160–167.
Sevryuk, M. B. (1991b). New results in the reversible KAM theory. In Lazutkin, V. F. (ed.),Proc. Workshop Dynamical Systems, Euler Intern. Math. Inst. St. Petersburg, Birkhäuser-Verlag.
Sevryuk, M. B. (1992). Reversible linear systems and their versal deformations.J. Sov. Math. 60(5), 1663–1680.
Sevryuk, M. B. (1993). New cases of quasi-periodic motion in reversible systems.Chaos 3(2), 211–214.
Xiu, J., You, J., and Qiu, Q. (1994). Invariant tori for nearly integrable Hamiltonian systems with degeneracy. Preprint ETH-Zürich.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Broer, H.W., Huitema, G.B. Unfoldings of quasi-periodic tori in reversible systems. J Dyn Diff Equat 7, 191–212 (1995). https://doi.org/10.1007/BF02218818
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02218818