Global C∞ Integrability of Cubic–Linear Polynomial Differential Systems

Yangyou Pan ^[1] ; Cong Wang ^[2] ; Xiang Zhang ^[2]
1. [1] Chizhou University
  
  Chizhou University
  
  China
2. [2] Shanghai Jiao Tong University
  
  Shanghai Jiao Tong University
  
  China
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 13, Nº 1, 2014, págs. 73-87
Idioma: inglés
DOI: 10.1007/s12346-013-0106-4
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Resumen
- The cubic–linear polynomial differential systems having at least one finite singularity are affine equivalent to the systems of the form x = P(x, y) = bx + cy + dx2 + exy + f y2 + gx3 + hx2 y + ixy2 + jy3, y = Q(x, y), with g2 + h2 + i 2 + j 2 = 0 (otherwise it is quadratic–linear), and Q(x, y) is either x or y. In this paper we classify all the cubic–linear systems with Q(x, y) = y which have a global C∞ first integral. Meanwhile we obtain some partial results on the existence of global analytic first integrals. For proving our results we will use the local characterization of first integrals, partition of unity in R2, smoothness of first integrals in canonical regions and so on.