zbMATH — the first resource for mathematics

Normal forms for local families and nonlocal bifurcations. (English) Zbl 0811.58051
Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 233-258 (1994).
This paper is a brief summary of the results and problems concerning the conjecture, which can be formulated as follows. Consider the family of differential equations (1) \(\dot x = \vec {v}(x, \varepsilon)\) on \(S^ 2\), \(\varepsilon \in B \subset \mathbb{R}^ k\). A polycycle is a finite union of singular points and continual phase curves of (1), which is connected and cannot be contracted along itself to any proper subset. A limit cycle is generated by a polycycle \(\gamma\) in the family (1) if the path \(\varepsilon(s)\) in the parameter space exists such that for any \(s \in (0, 1]\) the equation corresponding to \(\varepsilon (s)\) has a limit cycle \(\ell(s)\), continuously depending on the parameter \(s\), and \(\ell(s) \to \gamma\) as \(s \to 0\) in sense of the Hausdorff metric. Cyclicity of the polycycle in the family (1) is the maximal number of limit cycles generated by this polycycle and corresponding to the parameter value, close to critical one. Then the conjecture in question is: Cyclicity of any polycycle appearing in the typical finite parameter family is finite. The last conjecture implies the Hilbert-Arnold one.
This interesting paper contains unfortunately many misprints.
For the entire collection see [Zbl 0797.00019].
Reviewer: A.Klíč (Praha)

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
37G05 Normal forms for dynamical systems
Full Text: Link