Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Upper, down, two-sided Lorenz attractor, collisions, merging and switching

Diego Barros, Christian Bonatti, Maria Jose Pacifico

Abstract

We present a slightly modified version of the well known "geometric Lorenz attractor". It consists in a C1 open set O of vector fields in R3 having an attracting region U containing: (1) a unique singular saddle point sigma; (2) a unique attractor Lambda containing the singular point; (3) the maximal invariant in U contains at most 2 chain recurrence classes, which are Lambda and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along the union of 2 co-dimension 1 sub-manifolds which divide O in 3 regions. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expel the singular point sigma and becomes a horseshoe and the horseshoe absorbs sigma becoming a Lorenz attractor. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expel the singular point sigma and becomes a horseshoe and the horseshoe absorbs sigma becoming a Lorenz attractor.

Upper, down, two-sided Lorenz attractor, collisions, merging and switching

Abstract

We present a slightly modified version of the well known "geometric Lorenz attractor". It consists in a C1 open set O of vector fields in R3 having an attracting region U containing: (1) a unique singular saddle point sigma; (2) a unique attractor Lambda containing the singular point; (3) the maximal invariant in U contains at most 2 chain recurrence classes, which are Lambda and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along the union of 2 co-dimension 1 sub-manifolds which divide O in 3 regions. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expel the singular point sigma and becomes a horseshoe and the horseshoe absorbs sigma becoming a Lorenz attractor. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expel the singular point sigma and becomes a horseshoe and the horseshoe absorbs sigma becoming a Lorenz attractor.

Paper Structure

This paper contains 46 sections, 41 theorems, 23 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic topological notions
  4. Singular points
  5. The shift map $\mathfrak{S}$
  6. Hyperbolic notions
  7. Invariant manifold and foliations
  8. Cone fields
  9. Geometric Lorenz attractor
  10. An open set of singular hyperbolic flows
  11. A geometric model for $X \in {\mathcal{O}}_1$
  12. Singular hyperbolic conditions
  13. $X\in {\mathcal{O}}_1$ can be constructed in any $3$-dimensional manifold
  14. Topological dynamics: the attractor and the chain recurrence classes
  15. Quasi-attractor
  16. ...and 31 more sections

Key Result

Theorem A

Any vector field $X\in{\mathcal{L}}^+$ admits exactly $2$ chain recurrence classes: one is an upper-Lorenz attractor, and the other is a hyperbolic basic set, topologically equivalent to the suspension of a fake horseshoe. The symmetric statement holds for ${\mathcal{L}}^-$, interchanging the upper

Figures (22)

  • Figure 1: The diagram displays $\Sigma = \Sigma^1 \cup \Sigma^2$ and $W^s(\sigma) \setminus W^{ss}(\sigma)=W^s_{+}(\sigma) \cup W^s_{-}(\sigma)$, and the points $q_i, 1\leq i \leq 2,$ defined above.
  • Figure 2: The usual horseshoe and a fake horseshoe
  • Figure 3: $\Sigma_+$ and $\Sigma_-$: determined by $W_1$ and $W_2$, the stable manifolds of $p_1$ and $p_2$ respectivelly.
  • Figure 4: The bifurcation diagram of flows in $\mathcal{O}_{\varphi}$.
  • Figure 5: A geometric Lorenz flow, projection on $I$ through the stable leaves and a sketch of the image of one leaf under the return map, and the 1-dimensional map defined in the space of leaves of ${\mathcal{F}}^s$.
  • ...and 17 more figures

Theorems & Definitions (87)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Theorem H
  • Definition 2.1
  • Definition 2.2
  • ...and 77 more