Global topological classification of Lotka-Volterra quadratic differential systems

Schlomiuk, Dana ; Vulpe, Nicolae

Electronic Journal of Differential Equations, Tome 2012 (2012).

Résumé

Summary: The Lotka-Volterra planar quadratic differential systems have numerous applications but the global study of this class proved to be a challenge difficult to handle. Indeed, the four attempts to classify them (Reyn (1987), Wöz-Buserkros (1993), Georgescu (2007) and Cao and Jiang (2008)) produced results which are not in agreement. The lack of adequate global classification tools for the large number of phase portraits encountered, explains this situation. All Lotka-Volterra systems possess invariant straight lines, each with its own multiplicity. In this article we use as a global classification tool for Lotka-Volterra systems the concept of configuration of invariant lines (including the line at infinity). The class splits according to the types of configurations in smaller subclasses which makes it easier to have a good control over the phase portraits in each subclass. At the same time the classification becomes more transparent and easier to grasp. We obtain a total of 112 topologically distinct phase portraits: 60 of them with exactly three invariant lines, all simple; 27 portraits with invariant lines with total multiplicity at least four; 5 with the line at infinity filled up with singularities; 20 phase portraits of degenerate systems. We also make a thorough analysis of the results in the paper of Cao and Jiang [13]. In contrast to the results on the classification in [13], done in terms of inequalities on the coefficients of normal forms, we construct invariant criteria for distinguishing these portraits in the whole parameter space $\mathbb{R}^{12}$ of coefficients.

Détail

Classification : 58K30, 34A26, 34C05, 34C40
Keywords: quadratic vector fields, Lotka-Volterra differential systems, phase portraits, affine invariant polynomials, topological invariants