The dynamics of perturbations of the contracting Lorenz attractor.

*(English)*Zbl 0797.58051The 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 \(\lambda_ 2 < \lambda_ 3 < 0 < \lambda_ 1\), \(\lambda_ 3 + \lambda_ 1 > 0\).

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 |

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\textit{A. Rovella}, Bol. Soc. Bras. Mat., Nova Sér. 24, No. 2, 233--259 (1993; Zbl 0797.58051)

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