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Zbl 0801.58010
Bamón, Rodrigo; Labarca, Rafael; Mañé, Ricardo; Pacífico, José
The explosion of singular cycles. (English)
[J] Publ. Math., Inst. Hautes Étud. Sci. 78, 207-232 (1993). ISSN 0073-8301; ISSN 1618-1913/e

In the work of {\it S. Newhouse} and {\it J. Palis} [``Bifurcations of Morse-Smale dynamical systems'', Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 303-366 (1973; Zbl 0279.58011)] one-parameter families of diffeomorphisms corresponding to values of the parameter close to the first bifurcation parameter (i.e. the first value of the parameter for which the diffeomorphism is not Morse-Smale) were considered. In the article under review, as in the work of {\it S. Newhouse}, {\it J. Palis} and {\it F. Takens} [Publ. Math., Inst. Hautes Étud. Sci. 57, 5-71 (1983; Zbl 0518.58031)] the authors describe how the cycle explodes when the parameter increases. Explosion means a sudden increase of the size of a relevant dynamically defined set triggered by a small perturbation of the system. So, for example, in the above indicated works and in the paper of {\it S. Newhouse} and {\it J. Palis} in Astérisque, 31, 44-140 (1976; Zbl 0322.58009)] a perturbation of the system leads to the creation of homoclinic tangencies and then to the vast array of phenomena they carry on their wake (Newhouse wild horseshoes, persistent tangencies, non-hyperbolic attractors). The explosion of singular cycles (cycles containing a hyperbolic singularity) is also considered. The authors describe how they explode in a way entirely different from that of the cycles of diffeomorphisms of surfaces of the Afrajmovich-Shil'nikov cycles [{\it V. S. Afrajmovich} and {\it L. P. Shil'nikov}, Math. USSR, Izv. 8 (1974), 1235-1270 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1248-1288 (1974; Zbl 0322.58007)].
[B.V.Loginov (Ulyanovsk)]

MSC 2000:: *58E07 Abstract bifurcation theory
37G15 Bifurcations of limit cycles and periodic orbits
37G99 Bifurcation theory

Keywords: depending on a parameter; vector fields; singular cycles

Citations: Zbl 0322.58009; Zbl 0279.58011; Zbl 0518.58031; Zbl 0322.58007

Cited in: Zbl 1036.37018 Zbl 0903.34024

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