The explosion of singular cycles
type: Item
instance of: scholarly article
title: The explosion of singular cycles
zbMATH Open document ID: 0801.58010
DOI: 10.1007/BF02712919
author: Rodrigo Bamón
author: R. Labarca
author: Ricardo Mañé
author: José Pacífico
published in: Publications Mathématiques
publication date: 1994-11-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104092
review text: In the work of \textit{S. Newhouse} and \textit{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 \textit{S. Newhouse}, \textit{J. Palis} and \textit{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 \textit{S. Newhouse} and \textit{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 [\textit{V. S. Afrajmovich} and \textit{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)].
Mathematics Subject Classification ID: 58E07
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 569543
zbMATH Keywords: depending on a parameter
zbMATH Keywords: vector fields
zbMATH Keywords: singular cycles
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
cites work: ON ATTAINABLE TRANSITIONS FROM MORSE-SMALE SYSTEMS TO SYSTEMS WITH MANY PERIODIC MOTIONS
cites work: Structural stability of Lorenz attractors
cites work: Stability of singular horseshoes
cites work: Q3814426
cites work: Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms
cites work: On the Structure of Local Homeomorphisms of Euclidean n-Space, II
OpenAlex ID: W1963934010
description: scientific article; zbMATH DE number 569543