Chenciner, A. Bifurcations de diffĂ©omorphismes de \({\mathbb{R}}^ 2\) au voisinage d’un point fixe elliptique. (French) Zbl 0582.58025 Chaotic behaviour of deterministic systems, Les Houches/Fr. 1981, Sess. 36, 273-348 (1983). [For the entire collection see Zbl 0546.00032.] Starting with the classical Van Der Pol limit cycle, this course introduces, in the case of plane diffeomorphisms, some of the tools used in the local theory of bifurcations of elliptic fixed points: after an introductory chapter dealing with different ways of proving the persistence under perturbation of a fixed point, the so-called ”normal forms” are introduced and the resonances discussed. The structure of the proof of the classical Hopf bifurcation theorem is then explained, as well as the dynamics on the invariant curves. The last part of the course is an introduction to (an early stage of) some of the author’s work on degenerate (i.e. transcritical) Hopf bifurcations, where small denominators and Hamiltonian-like behaviour come into play [see ”Bifurcations de points fixes elliptiques”: I, Publ. Math., Inst. Hautes Etud. Sci. 61, 67-127 (1985; Zbl 0566.58025); II. Invent. Math. 80, 81- 106 (1985; Zbl 0578.58031); III, Preprint, Univ. Paris VII (1986)]. Cited in 6 Documents MSC: 37G99 Local and nonlocal bifurcation theory for dynamical systems Keywords:dynamical systems; invariant curves; diffeomorphisms; bifurcations Citations:Zbl 0546.00032; Zbl 0566.58025; Zbl 0578.58031 PDFBibTeX XML