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The explosion of singular cycles. (English) Zbl 0801.58010

In the work of S. Newhouse and 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 S. Newhouse, J. Palis and 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 S. Newhouse and 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 [V. S. Afrajmovich and 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)].

MSC:

58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
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References:

[1] V. S. Afraimovitch andL. P. Sil’nikov, On attainable transitions from Morse-Smale systems to systems with many periodic motions,Math. USSR Izvestija,8 (1974), 1235–1270. · Zbl 0322.58007 · doi:10.1070/IM1974v008n06ABEH002146
[2] J. Guckenheimer andR. Williams, Structural stability of Lorenz attractors,Publ. Math. IHES,50 (1979), 59–72. · Zbl 0436.58018
[3] R. Labarca andM. J. Pacífico, Stability of singular horseshoes,Topology,25 (1986), 337–352. · Zbl 0611.58033 · doi:10.1016/0040-9383(86)90048-0
[4] J. Milnor andW. Thurston,On iterated maps on the interval, I, II, Preprints, Princeton, 1977. · Zbl 0664.58015
[5] S. Newhouse andJ. Palis, Bifurcations of Morse-Smale Dynamical Systems,Dynamical Systems, edited byM. M. Peixoto, Academic Press, 303–366, 1973.
[6] S. Newhouse andJ. Palis, Cycles and bifurcation theory,Astérisque,31 (1976), 43–140.
[7] S. Newhouse, J. Palis andF. Takens, Bifurcations and stability of families of diffeomorphisms,Publ. Math. IHES,57 (1983), 5–72.
[8] J. Palis andF. Takens, Cycles and measure of bifurcation sets for two dimensional diffeomorphismsInventiones Math.,82 (1985), 397–422. · Zbl 0579.58005 · doi:10.1007/BF01388862
[9] M. J. Pacífico andA. Rovella, Contracting singular cycles,Ann. Ec. Norm. Sup., to appear.
[10] S. Sternberg, On the structure of local homeomorphisms of euclideann-space II,Am. J. Math.,80 (1958), 623–631. · Zbl 0083.31406 · doi:10.2307/2372774
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