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)


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


Zbl 0436.58018
Full Text: DOI


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