×

Abundance of strange attractors. (English) Zbl 0815.58016

Consider a \(C^ \infty\) one-parameter family of surface diffeomorphisms \((f_ \varepsilon)_ \varepsilon\) and suppose that \(f_ 0\) has a homoclinic tangency associated to some periodic point \(p_ 0\). The authors show that under some open and dense assumptions there exists a positive Lebesgue measure set \(E\) of parameter values near 0, such that for \(\varepsilon \in E\), \(f_ \varepsilon\) exhibits a strange attractor, or repellor, near the orbit of tangency.
The proof is based on the observation that near the tangency the family \(f_ \varepsilon\) unfolds to a family \(\phi_ \varepsilon\) of so-called Hénon-like maps, such as small perturbations of \((x,y) \mapsto (1 - ax^ 2 + \varepsilon y,\varepsilon x)\). To such families the authors extend and generalize the famous result of Benedicks and Carleson about the existence of strange attractors in the Hénon map. This is the main part of the paper, and it presents a very careful, complete and readable account of the intricate iterative construction required to prove the strangeness of the attractor (whose existence is not a big issue). Also, some aspects of the original proof are simplified.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] [BC1]Benedicks, M. &Carleson, L., On iterations of 1x2 on (, 1.).Ann. of Math., 122 (1985), 1–25. · Zbl 0597.58016 · doi:10.2307/1971367
[2] [BC2]–, The dynamics of the Hénon Map.Ann. of Math., 133 (1991) 73–169. · Zbl 0724.58042 · doi:10.2307/2944326
[3] [He]Hénon, M., A two dimensional mapping with a strange attractor.Comm. Math. Phys., 50 (1976), 69–77. · Zbl 0576.58018 · doi:10.1007/BF01608556
[4] [Py]Plykin, R., On the geometry of hyperbolic attractors of smooth cascades.Russian Math. Surveys, 39 (1984), 85–131. · Zbl 0584.58038 · doi:10.1070/RM1984v039n06ABEH003182
[5] [PT1]Palis, J. &Takens, F., Hyperbolicity and the creation of homoclinic orbits.Ann. of Math., 125 (1987), 337–374. · Zbl 0641.58029 · doi:10.2307/1971313
[6] [PT2]Palis, J. & Takens, F.,Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Fractal Dimensions and Infinitely Many Attractors. Cambridge University Press, 1990. · Zbl 0790.58014
[7] [Ru]Ruelle, D.,Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, 1989. · Zbl 0684.58001
[8] [RE]Ruelle, D. &Eckmann, J., Ergodic theory of chaos and strange attractors.Rev. Modern Phys., 57 (1985), 617–656. · Zbl 0989.37516 · doi:10.1103/RevModPhys.57.617
[9] [Sh]Shub, M.,Global Stability of Dynamical Systems. Springer-Verlag, 1987. · Zbl 0606.58003
[10] [Si]Singer, D., Stable orbits and bifurcations of maps of the interval.SIAM J. Appl. Math., 35 (1978), 260. · Zbl 0391.58014 · doi:10.1137/0135020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.