Bifurcations de points fixes elliptiques. III: Orbites périodiques de “petites” périodes et élimination résonnante des couples de courbes invariantes. (Bifurcations of elliptic fixed points. III: Periodic orbits with “small” periods and resonant elimination of pairs of invariant curves.). (French) Zbl 0712.58044

Chenciner, Alain

Publ. Math., Inst. Hautes Étud. Sci. 66, 5-91 (1988).

The paper is the third part of a general study devoted to generic two parameter families of smooth local diffeomorphism of \({\mathbb{R}}^ 2\), written as perturbations of normal forms. [For part I see ibid. 61, 67- 127 (1985; Zbl 0566.58025), for part II see Invent. Math. 80, 81-106 (1985; Zbl 0578.58031)].
The first paper was concerned by invariant sets, the rotation number of which is a “good” irrational number, in the sense that the diffeomorphism has an invariant curve, close to a circle on which the induced map is smoothly conjugate to the rotation. In a neighbourhood of the rotation center, it has the same singularities as those of the normal form. This paper deals with the case of invariant sets with “good” rational rotation number, defined equivalently in its first section. Every periodic orbit with a good rotation number is well ordered.
In the second section, from the considered generic situation, these periodic orbits are interpreted as the “souvenir of nearly resonances”, being obtained by bifurcation from resonant case. Some restriction of the diffeomorphism to a ring appears as a perturbation of a resonant normal form invariant by the simple rotation. An interpolation by a two parameters family of differential equations is considered, the study of which is transferred in an appendix.
The bifurcation diagram in the parameter plane makes appear infinitely many “bubbles” arranged in a string along a curve related to the merging of two invariant curves for the family of normal forms. In the complementary part of these bubbles the diffeomorphism behaves like a normal form. The third and fourth sections show that the bubbles become very narrow near points corresponding to good rational rotation numbers.
The last section is devoted to complex phenomena occurring for some parameter values belonging to resonant tongues. The diffeomorphism has an ordered hyperbolic periodic orbit with homoclinic points, not lying on an invariant curve, in an equivalent situation of the saddle-node invariant curve of the corresponding normal form.
This paper constitutes an important contribution to the knowledge of such two parameter families of diffeomorphisms, with in particular the description of the chain of infinitely many bubbles and its properties. However it would be more informative to complete the references by those giving more ancient results on one parameter families when a bubble is crossed. Such results are related to “exceptional” resonant bifurcation cases, deduced from Cigala normal forms. In particular, figures 20 were described in 1973, 1980 for rotation numbers 1/3, 1/4.

Reviewer: C.Mira

MSC:

37G99	Local and nonlocal bifurcation theory for dynamical systems
37G05	Normal forms for dynamical systems
57R50	Differential topological aspects of diffeomorphisms
70K40	Forced motions for nonlinear problems in mechanics

Keywords:

resonance; map; iteration; invariant sets; rotation number; diffeomorphism; periodic orbit; bifurcation

Citations:

Zbl 0566.58025; Zbl 0578.58031

References:

[1]	V. I. Arnold,Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir, 1980.
[2]	D. G. Aronson, M. A. Chory, G. R. Hall, R. P. McGehee, Bifurcations from an invariant circle for 2-parameter families of maps of the plane: a computer assisted study,Commun. Math. Phys.,83 (1983), 303–354. · Zbl 0499.70034
[3]	G. D. Birkhoff, Sur certaines courbes fermées remarquables,Bull. Soc. Math. de France,60 (1932), 1–26. · JFM 58.0633.01
[4]	P. Boyland etG. R. Hall, Invariant circles and the order structure of periodic orbits in monotone twist maps,Topology, à paraître. · Zbl 0618.58032
[5]	H. Broer etF. Takens,Formally symmetric normal forms and genericity, Prépublication Université de Groningen, 1986.
[6]	A. Chenciner, Points homoclines au voisinage d’une bifurcation de Hopf dégénérée de difféomorphismes deR 2,C.R.A.S., Série I,294 (1982), 269–272. · Zbl 0509.58014
[7]	A. Chenciner, Bifurcations de difféomorphismes deR 2 au voisinage d’un point fixe elliptique, inLes Houches Session XXXVI, 1981,Comportement chaotique des systèmes déterministes, Iooss, Helleman, Stora édit., North Holland, 1983, 273–348.
[8]	A. Chenciner, Bifurcations de points fixes elliptiques. I: Courbes invariantes,Publications Math. de l’I.H.E.S.,61 (1985), 67–127. · Zbl 0566.58025
[9]	A. Chenciner, Bifurcations de points fixes elliptiques. II: Orbites périodiques et ensembles de Cantor invariants,Inventiones Math.,80 (1985), 81–106. · Zbl 0578.58031
[10]	A. Chenciner, Resonant elimination of a couple of invariant closed curves in the neighborhood of a degenerate Hopf bifurcation of diffeomorphisms ofR 2,IIASA Workshop on dynamic processes, Sopron, septembre 1985,Springer L.N. in Economics and Mathematical Systems,287 (1987), 3–9.
[11]	A. Chenciner, A. Gasull, etJ. Llibre, Une description complète du portrait de phase d’un modèle d’élimination résonnante,C.R.A.S., Serie I,305 (1987), 623–626. · Zbl 0647.34042
[12]	F. Dumortier, P. R. Rodriguez, R. Roussarie, Germs of diffeomorphisms in the plane,Springer L.N.,902, 1981. · Zbl 0502.58001
[13]	K. Hockett, P. Holmes, Josephon’s junction, annulus maps, Birkhoff attractors, Horseshoes and rotation sets,Ergodic theory and dynamical systems,6 (1986), 205–239. · Zbl 0593.58020
[14]	G. Iooss,Bifurcation of maps and applications, North Holland, 1979. · Zbl 0408.58019
[15]	J. R. Johnson,Some properties of a 3-parameter family of diffeomorphisms of the plane near a transcritical Hopf bifurcation, Thesis, University of Minnesota, July 1985.
[16]	P. Lecalvez, Existence d’orbites quasi-périodiques dans les attracteurs de Birkhoff,Commun. Math. Phys.,106 (1986), 383–394. · Zbl 0602.58031
[17]	HansJ. A. Metz, HansLauwerier,The dynamical behaviour of some parasitoïd-host model : k-cycles and a transcritical Hopf bifurcation in a planar difference equation, Preprint, juin 1984.
[18]	J. Moser, Non existence of integrals for canonical systems of differential equations, C.P.A.M.,8 (1955), 409–436. · Zbl 0068.29402
[19]	J. Moser, Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff, inGeometry and Topology, Rio de Janeiro, 1976.Springer L.N.,597 (1977), 464–494.
[20]	J. Moser, Lectures on Hamiltonian systems,Memoirs of the A.M.S.,81 (1968), 1–60. · Zbl 0172.11401
[21]	E. Zehnder, Homoclinic points near elliptic fixed points,C.P.A.M.,26 (1973), 131–182. · Zbl 0261.58002