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Periodic orbits for dissipative twist maps

Published online by Cambridge University Press:  19 September 2008

Martin Casdagli
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA and Mathematics Institute, University of Warwick, Coventry CV47AL, England
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Abstract

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We develop simple topological criteria for the existence of periodic orbits in maps of the annulus. These are applied to one-parameter families of dissipative twist maps of the annulus and their attractors. It follows that many of the motions found by variational methods in area preserving twist maps also occur in the dissipative case.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[1]Arnold, V. I. & Avez, A.. Ergodic Problems of Classical Mechanics, pp. 8890. Benjamin: New York 1968.Google Scholar
[2]Aronson, D. G., Chory, M. A., Hall, G. R. & McGehee, R. P.. Bifucations from an invariant circle for two-parameter families of maps of the plane: A computer assisted study. Commun. Math. Phys. 83 (1982), 303354.CrossRefGoogle Scholar
[3]Aubry, S. & Le Daeron, P. L.. The discrete Frenkel-Kontorova model and its extensions. Physica 8D (1983), 381422.Google Scholar
[4]Birkhoff, G. D.. On the periodic motions of dynamical systems. Acta Mathematica 50, 359379.CrossRefGoogle Scholar
Reprinted in Collected Mathematical Papers, Vol. II, American Math. Soc, New York 1950.Google Scholar
[5]Birkhoff, G. D.. Sur quelques courbes fermées remarkables. Bull. Soc. Math. de France, 60 (1932), 126.Google Scholar
Reprinted in Collected Mathematical Papers, Vol. II, American Math. Soc, New York 1950.Google Scholar
[6]Bohr, T.. A bound for the existence of invariant circles in a class of two-dimensional dissipative maps. Preprint, Cornell University (1984).CrossRefGoogle Scholar
[7]Bohr, T., Bak, P. & Jensen, M. H.. Transition to chaos by interaction of resonances in dissipativesystems. Preprint, Cornell University (1984).Google Scholar
[8]Calvez, P. Le. Existence d'orbites quasi-periodiques dans les attracteurs de Birkhoff. Preprint, Université Paris, Orsay (1985).Google Scholar
[9]Calvez, P. Le. Private communication.Google Scholar
[10]Chenciner, A.. Sur un énoncé dissipative du théorèm géometrique de Poincare-Birkhoff. C.R.A.S. t294, Série I (1982), 243245.Google Scholar
[11]Chenciner, A.. Orbites périodiques et ensembles de Cantor invariant d' Aubry-Mather au voisinage d'une bifurcation de Hopf dégénérée de difféomorphismes de ℝ2. C.R.A.S. t297, Série I (1983), 465467.Google Scholar
[12]Hall, G. R.. A topological version of a theorem of Mather on twist maps. Preprint, University of Wisconsin, Madison (1983).CrossRefGoogle Scholar
[13]Herman, M.. Sur les courbes invariantes par les difféomorphismes de l'anneau. Vol. I, Ch. 1. Astérisque, 103104 (1983).Google Scholar
[14]Hockett, K. & Holmes, P.. Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets. Ergod. Th. & Dynam. Sys. 6 (1986), 205239.CrossRefGoogle Scholar
[15]Hocking, J. G. & Young, G. S.. Topology. Addison-Wesley: Reading 1961.Google Scholar
[16]Ito, R.. Rotation sets are closed. Math. Proc. Comb. Phil. Soc. 89 (1981), 107111.CrossRefGoogle Scholar
[17]Katok, A.. Some remarks on Birkhoff and Mather twist map theorems. Ergod. Th. & Dynam. Sys. 2 (1982), 185194.CrossRefGoogle Scholar
[18]Katok, A.. Periodic and quasiperiodic orbits for twist maps. In Proceedings, Sitges 1982 Garrido, L. (ed.), Springer-Verlag, Berlin.Google Scholar
[19]Mather, J. N.. Existence of quasiperiodic orbits for twist homomorphisms. Topology 21 (1982), 457467.CrossRefGoogle Scholar
[20]Nitecki, Z.. Differentiable Dynamics. M.I.T. Press: 1971.Google Scholar
[21]Ostlund, S., Rand, D., Sethna, J. & Siggia, E.. Universal properties of the transition from quasiperiodicity to chaos in dissipative systems. Physica 8D (1983), 303342.Google Scholar