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Bifurcation of an attracting invariant circle: a Denjoy attractor

Published online by Cambridge University Press:  19 September 2008

Glen R. Hall
Glen R. Hall
Affiliation:
Mathematics Research Center, 610 Walnut Street, University of Wisconsin, Madison, Wisconsin, 53705
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Abstract

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We construct an example of a C∞ diffeomorphism of an annulus into itself which has an attracting invariant circle such that the map restricted to this circle has no periodic points and no dense orbits. By studying two parameter families of maps of the plane which undergo Hopf bifurcation, particularly the set of parameter values for which the rotation number is irrational, we see that the above example can be considered as a ‘worst case’ of the loss of smoothness of an attracting invariant circle without periodic orbits.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 1 , March 1983 , pp. 87 - 118
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Arnold, V. I.. Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields. Functional Analysis and its Applications 11 (2) (1977), 1–10.Google Scholar
[2]Aronson, D. G., Chory, M. A., Hall, G. R. & McGehee, R. P.. Bifurcation from an invariant circle for two parameter families of maps of the plane: a computer-assisted study. Comm. Math. Phys. 83 (1982), 303–354.CrossRefGoogle Scholar
[3]Bushard, L. B.. Periodic solutions and locking-in on the periodic surface. J. Non-linear Mechanics 8 (1973), 129–141.CrossRefGoogle Scholar
[4]Conley, C.. Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics. No. 38. American Mathematical Society: Providence, 1978.CrossRefGoogle Scholar
[5]Curry, J. & Yorke, J.. A transition from Hopf bifurcation to chaos: computer assisted experiments on maps in ℝ2. The Structure of Attractors in Dynamical Systems (ed. Markley, N. G., Martin, J. C., Perrizo, W.), Lecture Notes in Mathematics 668, pp 48–68. Springer-Verlag: Berlin, 1978.CrossRefGoogle Scholar
[6]Denjoy, A.. Sur les courbes definies par les equations differentialles a la surface du tore, J. de Math. Pures et Appliquees, 11 (1932), 33–375.Google Scholar
[7]Fenichel, N.. Persistance and smoothness of invariant manifolds for flows. Indiana University Mathematics Journal 21 (3) (1971), 193–226.CrossRefGoogle Scholar
[8]Fenichel, N.. The orbit structure of the Hopf bifurcation problem. J. Diff. Eq. 11 (2) (1975), 308–328.CrossRefGoogle Scholar
[9]Handel, M.. A pathological area preserving C ∞ diffeomorphism of the plane. Proc. Amer. Math. Soc. (To appear.)Google Scholar
[10]Harrison, J.. Denjoy fractals in the plane. Preprint.Google Scholar
[11]Harrison, J.. A C 2 counter example to the Seifert conjecture. Preprint.Google Scholar
[12]Herman, M. R.. Sur la conjugaison differentialles des diffeomorphisms du circle a des rotation. Publ. Math. IHES 49 (1979), 5–233.CrossRefGoogle Scholar
[13]Herman, M. R.. Contre-exemple de Denjoy et contre-exemples de class C 3−ε au theoreme des courbes invariantes un nombre de rotation fixe. (To appear as chapter III of Introduction a l'etude des courbes invariante des diffeomorphism de l'annneau.)Google Scholar
[14]Hirsch, M. W., Pugh, C. & Shub, M.. Invariant Manifolds. Lecture Notes in Mathematics 583. Springer-Verlag: Berlin, 1977.CrossRefGoogle Scholar
[15]Knill, R. J.. A C ∞ flow on S 3 with a Denjoy minimal set. J. Diff. Geom. 16 (1982), 271–280.Google Scholar
[16]Marsden, J. & McCracken, M.. The Hopf Bifurcation and its Applications. Notes in Applied Science 19, Springer-Verlag: Berlin, 1976.CrossRefGoogle Scholar
[17]McGehee, R. P.. The stable manifold theorem via an isolating block. Symposium on Ordinary Differential Equations (ed. Harris, W. A., Sibuya, Y.), Lecture Notes in Mathematics 312, pp 135–144. Springer-Verlag: Berlin, 1972.Google Scholar
[18]Neimark, M.. On some cases of dependence of periodic motion on parameters. Dokl. Akad. Nauk. SSSR 129 (1959), 736–739.Google Scholar
[19]Poincare, H.. Les Methods Nouvelles de la Mechanic Celeste Vol. 3. Gauthiers-Verlag, 1899.Google Scholar
[20]Ruelle, D. & Takens, F.. On the nature of turbulence. Comm. Math. Phys. 20 (1971), 126–137.CrossRefGoogle Scholar
[21]Sacker, R.. On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations. Thesis, New York University, IMM-NYU (10 1964).Google Scholar
[22]Selgrade, J. F.. Isolated invariant sets for flows on vector bundles. Trans. Amer. Math. Soc. 203 (1975), 359–390.CrossRefGoogle Scholar