Valery Gaiko: Global Qualitative Analysis of Polynomial Dynamical Systems and Hilbert’s Sixteenth Problem

We carry out the global qualitative analysis of planar polynomial dynamical systems and suggest a new bifurcational 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 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 a solution of Hilbert’s Sixteenth Problem on the maximum number of limit cycles surrounding a unique singular point for arbitrary polynomial systems and solve the limit cycle problem for a general Liénard polynomial system with an arbitrary (but finite) number of singular points. 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 neuronal cubic systems, generalized Lotka-Volterra quartic dynamical systems which are used as mathematical models of real biomedical and ecological systems. Finally, applying the same approach and using some numeric and analytic results, we consider three-dimensional polynomial dynamical systems and discuss how to complete the strange attractor bifurcation scenario in the classical Lorenz system globally connecting the homoclinic, period-doubling, Andronov-Shilnikov, and period-halving bifurcations of its limit cycles (Smale’s Fourteenth Problem).