Global topological classification of Lotka-Volterra quadratic differential systems

Electronic Journal of Differential Equations

关键词: Quadratic vector fields;    Lotka-Volterra differential systems;    phase portraits;    affine invariant polynomials;    topological invariants;   

DOI  :  

来源: DOAJ

【 摘 要 】

The Lotka-Volterra planar quadratic differential systems havenumerous applications but the global study of this class proved tobe a challenge difficult to handle. Indeed, the four attempts toclassify them (Reyn (1987), W"orz-Buserkros (1993),Georgescu (2007) and Cao and Jiang (2008)) produced results whichare not in agreement. The lack of adequate global classificationtools for the large number of phase portraits encountered,explains this situation. All Lotka-Volterra systems possessinvariant straight lines, each with its own multiplicity. In thisarticle we use as a global classification tool forLotka-Volterra systems the concept of configuration of invariantlines (including the line at infinity). The class splits accordingto the types of configurations in smaller subclasses which makesit easier to have a good control over the phase portraits in eachsubclass. At the same time the classification becomes moretransparent and easier to grasp. We obtain a total of 112topologically distinct phase portraits: 60 of them with exactlythree invariant lines, all simple; 27 portraits with invariantlines with total multiplicity at least four; 5 with the line atinfinity filled up with singularities; 20 phase portraits ofdegenerate systems. We also make a thorough analysis of theresults in the paper of Cao and Jiang [13]. In contrastto the results on the classification in [13], done interms of inequalities on the coefficients of normal forms, weconstruct invariant criteria for distinguishing these portraitsin the whole parameter space $mathbb{R}^{12}$ of coefficients.

【 授权许可】

Unknown