# The geometric lorenz attractor is a homoclinic class

The geometric lorenz attractor is a homoclinic class
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An attractor is a transitive set to which all nearby positiveorbits converge. An example of an attractor is the geometric Lorenz attractor GH. In this paper we prove that the geometric Lorenz attractor is a homoclinic class.

Tipo de documento: Artículo - Article

Palabras clave: Attractors, Flows, Periodic Orbits.

Source: http://www.bdigital.unal.edu.co

## Teaser

Boletı́n de Matemáticas
Nueva Serie, Volumen XI No.

1 (2004), pp.

69–78
THE GEOMETRIC LORENZ ATTRACTOR IS A
HOMOCLINIC CLASS
SERAFÍN BAUTISTA (*)
To Jairo Charris in memoriam.
Abstract.

An attractor is a transitive set to which all nearby positive
orbits converge.

An example of an attractor is the geometric Lorenz
attractor [GH].

In this paper we prove that the geometric Lorenz attractor
is a homoclinic class.
Key words and phrases.Attractors, Flows, Periodic Orbits.
2000 Mathematics Subject Classification.

Primary 37D45, Secondary
37C27.
1.

Introduction
The geometric Lorenz attractor and the horseshoe are two basic examples in the
modern theory of dynamical systems [PT].

The latter is a classical example of
a hyperbolic set and it was a main motivation to build up the hyperbolic theory
[PT].

The former was introduced in an attempt to model the Lorenz equation
[GW].

These two examples are different from the hyperbolic viewpoint: the
horseshoe is a hyperbolic set while the geometric Lorenz attractor is not.

Still,
the geometric Lorenz attractor resembles the horseshoe in some aspects: each
one of them is the closure of its periodic orbits, transitive (see Corollary 1) and
has sensitivity with respect to initial conditions.
A homoclinic class is the closure of the transverse intersection points of the
stable and unstable manifold of a hyperbolic periodic orbit.

In this paper we
show that the geometric Lorenz attractor is a homoclinic class.
2.

Basic definitions and Theorem 1
Hereafter M denotes a compact 3-manifold.

Let X be a vector field of class
C r , r ≥ 2.

We denote by Xt , t ∈ IR the flow generated by X.

Recall that this
flow is a C r action of IR into M , i.e., X : IR × M → M , where X0 = idM and
(*) Serafı́n Bautista.

Universidad Nacional de Colombia, Bogotá
E-email: sbautistad@unal.edu.co.
69
70
SERAFÍN BAUTISTA
Xs ◦ Xt = Xs t for all s, t ∈ IR.

An orbit of X is the set O = OX (q) = {Xt (q) :
t ∈ IR} for some q ∈ M .

The o...