Introduction

Joan C. Artés, Jaume Llibre and Alex C. Rezende
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Joan C. Artés: Universitat Autònoma de Barcelona, Departament de Matemàtiques
Jaume Llibre: Universitat Autònoma de Barcelona, Departament de Matemàtiques
Alex C. Rezende: Universitat Autònoma de Barcelona, Departament de Matemàtiques

Chapter Chapter 1 in Structurally Unstable Quadratic Vector Fields of Codimension One, 2018, pp 1-19 from Springer

Abstract: Abstract A vector field X : ℝ 2 → ℝ 2 $$X: {\mathbb R}^2 \to {\mathbb R}^2$$ of the form X = (P, Q) where P = ∑ a i j x i y j $$P= \sum a_{ij}x^iy^j$$ and Q = ∑ b i j x i y j $$Q= \sum b_{ij}x^iy^j$$ , 0 ≤ i + j ≤ n, is called a planar polynomial vector field of degree ≤ n. If ∑i+j=n(|a ij| + |b ij|) ≠ 0 then we say that X has degree n. In particular, polynomial vector fields of degree two are called quadratic vector fields. The M = (n + 1)(n + 2) real numbers a ij, b ij are called the coefficients of X. The space of these vector fields, endowed with the structure of an affine ℝ M $${\mathbb R}^M$$ -space in which X is identified with the M-tuple (a 00, a 10, …, a 0n, b 00, b 10, …, b 0n) of its coefficients, is denoted by P n ( ℝ 2 ) $${\textsl {P}}_n ({\mathbb R}^2)$$ .

Date: 2018
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DOI: 10.1007/978-3-319-92117-4_1

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