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), Wörz-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 R^{12} of coefficients.
Type of paper: Bifurcation diagram with global algebraic invariants.
Professors Schlomiuk and Vulpe have several papers classifying phase portraits according to the number and multiplicity of their invariant straight lines. Some of them classify the configurations of invariant straight lines, and others their phase portraits.
There are some few repetitions of phase portraits in this paper but this is due that the same topological phase portrait may appear with a different geometric set of invariant straight lines.
Even thought the authors claim in the abstract that they have found 112 topologically different portraits, the real number is 111. Since they do not give the detailed list of those which they find to be different, one cannot know which is the couple that they have interpreted to be the same.
The goal of this paper is very similar to the goal of Phase portraits of a quadratic system of differential equations occurring frequently in applications and they should have to contain the same phase portraits. But we find here some phase portraits that cannot be found in the other. We detail each difference.
Phase portrait \(QS145^{(5)}_1\) is present here, but it is not present in the other. Morever, its existence is confirmed by other papers. It seems to be an error in the paper or Reyn.
There are several degenerate phase portraits which are present in this paper but not in the other. Concretely \(QS176^{(4)}_1\), \(QS178^{(4)}_1\), \(QS180^{(5)}_1\), \(QS182^{(6)}_1\), \(QS183^{(5)}_1\), \(QS184^{(6)}_1\), \(QS186^{(6)}_1\), \(QS188^{(6)}_1\), \(QS194^{(7)}_1\), \(QS196^{(8)}_1\), \(QS198^{(6)}_1\) and \(QS200^{(7)}_1\). Possibly the normal forms considerend by Reyn did not cover all the possibilities of degenerate systems.