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Shape index, Brouwer degree and Poincaré-Hopf theorem

Héctor Barge, José M. R. Sanjurjo

Abstract

In this paper we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar% é-Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.

Shape index, Brouwer degree and Poincaré-Hopf theorem

Abstract

In this paper we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar% é-Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.
Paper Structure (4 sections, 14 theorems, 36 equations)

This paper contains 4 sections, 14 theorems, 36 equations.

Table of Contents

  1. Shape index, initial sections and the degree of a vector field
  2. Brouwer degree of vector fields near non-saddle sets
  3. Brouwer degree and connecting orbits in attractor repeller decompositions
  4. A generalitation of Borsuk's and Hirsch's antipodal theorems

Key Result

Theorem 1

Let $\varphi:\mathbb{R}^n\times\mathbb{R}\longrightarrow\mathbb{R}^n$ be a flow induced by a smooth vector field $F$ defined on $\mathbb{R}^n$. Suppose that $K$ is an isolated invariant set for $\varphi$ and $N$ an isolating block for $K$. Then, Moreover, if $K$ contains only a finite number of equilibria then

Theorems & Definitions (31)

  • Theorem 1
  • Definition 1.1
  • Theorem 1.2
  • proof
  • Remark 1.3
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • proof
  • Proposition 1.6
  • ...and 21 more