## The dynamics of perturbations of the contracting Lorenz attractor.(English)Zbl 0797.58051

The author considers the Lorenz equation where the eigenvalues $$\lambda_ 2 < \lambda_ 3 < 0 < \lambda_ 1$$ are modified by replacing the usual expanding condition $$\lambda_ 3+ \lambda_ 1 > 0$$ with a contracting condition $$\lambda_ 3 + \lambda_ 1 < 0$$. Denoting by $$X^ t$$ the flow of diffeomorphisms generated by a vector field $$X$$ and defining the notion of $$k$$-dimensionally almost persistent attractor, he proves the following result:
Theorem. There exists a $$C^ \infty$$ vector field $$X_ 0$$ in $$\mathbb{R}^ 3$$ having an attractor $$\Lambda$$ containing a singularity, and satisfying the following properties:
(a) There exist a local basin $$U$$ of $$\Lambda$$, a neighbourhood $$\mathcal U$$ of $$X_ 0$$, and an open and dense subset $${\mathcal U}_ 1$$ of $$\mathcal U$$, such that for all $$X \in {\mathcal U}_ 1$$, $$\Lambda_ X = \bigcap_{t \geq 0} X^ t(U)$$ consists of the union of one or at most two attracting periodic orbits, a hyperbolic set of topological dimension one, a singularity, and wandering orbits linking them.
(b) $$\Lambda$$ is 2-dimensionally almost persistent in the $$C^ 3$$ topology.
The paper is close to the paper of J. Guckenheimer and R. F. W. Williams [Publ. Math. Inst. Hautes Étud. Sci. 50, 59-72 (1979; Zbl 0436.58018)], where a $$C^ \infty$$ vector field $$X_ 0$$ on $$\mathbb{R}^ 3$$ is produced such that it has a $$C^ 1$$ persistent attractor $$\Lambda$$ containing a singularity with eigenvalues $$\lambda_ 2 < \lambda_ 3 < 0 < \lambda_ 1$$, $$\lambda_ 3 + \lambda_ 1 > 0$$.
Reviewer: J.Kalas (Brno)

### MSC:

 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 34D45 Attractors of solutions to ordinary differential equations 34D10 Perturbations of ordinary differential equations

### Keywords:

perturbations; persistence; Lorenz attractor

Zbl 0436.58018
Full Text:

### References:

 [1] [ACT] Arneodo, A., Coullet, P., Tresser, C. (1981),A. possible new mechanism for the onset of turbulence. Physics Letter, 81A/4. · Zbl 0485.58013 [2] [BC1] Benedicks, M., Carleson, L. (1985)On iterations of 1?ax 2 on (?1,1). Ann. Math., 122. · Zbl 0597.58016 [3] [BC2] Benedicks, M., Carleson, L. (1991),The dynamics of the Hénon map. Ann. Math., 133. · Zbl 0724.58042 [4] [G] Guckenheimer (1979),On sensitive dependence on initial conditions for one dimensional maps. Comm. in Math. Phys., 70. · Zbl 0429.58012 [5] [GW] Guckenheimer, Williams (1979),Structural stability of Lorenz attractors. Publ. Math. IHES, 50. · Zbl 0436.58018 [6] [HPS] Hirsch, Pugh, Schub (1977),Invariant manifold. Springer Lecture notes, 583. [7] [L] Lorenz (1963),Deterministic non-periodic flow. Journal of Atmosphere Sciences, 20. [8] [MV] Mora, L., Viana, M.,Abundance of strange attractors. To appear in Acta Mathematica. · Zbl 0815.58016 [9] [M1] Mañé, R. (1978),Persistent manifolds are normally hyperbolic. Trans. A.M.S., 246. [10] [M2] Mañé, R. (1985), (1987),Hyperbolicity, sinks and measure in one dimensional dynamics. Comm. in Math. Phys., 100 495-524 and Erratum-Comm. in Math. Phys., 112. · Zbl 0583.58016 [11] [R] Richlik, M. R. (1990),Lorenz attractor through Sil’nikov-type bifurcation. Part I. Erg. th. and Dyn. Syst., 10. [12] [S] Singer (1978),Stable orbits and bifurcations of maps of the internal. SIAM. J. Appl. Math., 35. · Zbl 0391.58014
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.