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Zbl 0857.58036
Labarca, Rafael
Bifurcation of contracting singular cycles. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 28, No. 6, 705-745 (1995). ISSN 0012-9593

Let $X$ be a $C^r$-vector field on $M^m$, where $M^m$ is a $C^\infty$, $m$-dimensional, compact, connected, boundaryless manifold. \par Roughly speaking, a cycle for the vector field $X$ is a compact invariant set $\Gamma\subset M$, $\Gamma=\Gamma_0\cup \Gamma_1$, where $\Gamma_0$ is formed by a finite number of singularities and periodic orbits of $X$, $\Gamma_0= \{\sigma_0,\dots,\sigma_n\}$, and $\Gamma_1$ consists of non-periodic regular trajectories of the vector field $X$, that ``connect'', by the prescribing way, the elements of $\Gamma_0$, or more precisely -- the $\alpha$-limit and $\omega$-limit sets of the trajectories from $\Gamma_1$ are subsets of the corresponding orbits from $\Gamma_0$. \par This paper deals with a 3-dimensional hyperbolic singular cycle $\Gamma\subset M^3$ that contains a unique hyperbolic singularity $\sigma_0$ and hyperbolic periodic orbits $\sigma_1,\dots,\sigma_n$ $(n\geq1)$ of the fector field $X$. \par The aim of this extended work is to continue the analysis of a new mechanism, the singular cycles, through which a vector field, depending on a parameter, may evolve when the parameter varies from a vector field exhibiting simple dynamics into one having nontrivial dynamics. Specifically, if we start with a Morse-Smale vector field and move through a generic one-parameter family of vector fields to a contracting singular cycle and beyond, we reach a region filled up mostly with hyperbolic flows. In fact, the Lebesgue measure of parameter values corresponding to non Axiom A flows is zero. Moreover, a complete description of the bifurcation sets that appear in these families is provided. \par This interesting paper is well written and organized.
[A.Klíč (Praha)]

MSC 2000:: *37G99 Bifurcation theory
37G15 Bifurcations of limit cycles and periodic orbits
37D15 Morse-Smale systems

Keywords: contracting singular cycle; first return map; foliation; bifurcation

Citations: Zbl 0694.65041

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