On Hilbert’s 16th Problem and Applications

We carry out the global qualitative analysis of planar polynomial dynamical systems and suggest a new geometric approach to solving Hilbert’s Sixteenth Problem on the maximum number and relative position of their limit cycles in two special cases of such systems. First, using geometric properties of four field rotation parameters of a new constructed canonical system, we present the proof of our earlier conjecture stating that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is (3:1). Then, by means of the same geometric approach, we solve the Problem for the classical Liénard polynomial system (in this special case, it is called Smale’s Thirteenth Problem). Besides, generalizing the obtained results, we present the solution of Hilbert’s Sixteenth Problem on the maximum number of limit cycles surrounding a singular point for arbitrary polynomial systems. Applying the Wintner-Perko termination principle for multiple limit cycles, we develop also an alternative approach to solving the Problem. By means of this approach, for instance, we complete the global qualitative analysis of Liénard-type cubic and piecewise linear dynamical systems, FitzHugh-Nagumo and Oja cubic systems, generalized Lotka-Volterra quartic dynamical systems which are used as mathematical models of real biomedical and ecological systems.