From local to global dynamics in Kolmogorov polynomial vector fields

Abstract

In this paper we provide an approach to characterize global dynamics from local linearized dynamics of Kolmogorov polynomial vector fields, and establish a link between the integrability of the vector field and the intersection number of the corresponding algebraic curves. Specially, a new criterion on nonexistence of limit cycles is given for Kolmogorov polynomial vector fields with any degree n. As an application of the results, we consider Kolmogorov quadratic and cubic polynomial vector fields, whose number of either center-type equilibria or weak saddles reaches the maximum in the interior of quadrants of real plane denoted by Int$R^2$, and obtain all topological classifications of their global dynamics in Poincaré disc by index theory and qualitative analysis. Notably, it is shown that the local dynamics of Kolmogorov quadratic polynomial vector fields (weakly nonlinear) having a center-type equilibrium or a weak saddle in Int$R^2$ can completely determine its global dynamics in Poincaré disc, but the local dynamics of Kolmogorov cubic polynomial vector fields (strongly nonlinear) having four center-type equilibria or four weak saddles in Int$R^2$ cannot completely determine its global dynamics in Poincaré disc.