Global Bifurcation Theory and Applications

of Polynomial Dynamical Systems


Valery A. Gaiko

National Academy of Sciences of Belarus


Multi-parameter polynomial dynamical systems are considered. We carry out a global bifurcation analysis and study applications of such systems. To control global bifurcations of limit cycles in planar systems, it is necessary to know the properties and combine the effects of all their rotation parameters. It can be done by means of the development of new bifurcation geometric methods based on the Wintner–Perko termination principle. Using these methods, we present, e. g., a solution of Hilbert’s Sixteenth Problem on the maximum number and distribution of limit cycles for the Kukles cubic-linear system, the general Liénard polynomial system with an arbitrary number of singular points, the Euler–Lagrange–Liénard polynomial mechanical system, Leslie–Gower systems which model the population dynamics in real ecological or biomedical systems, and a reduced planar quartic Topp system which models the dynamics of diabetes. Applying a similar approach, we study also three-dimensional polynomial dynamical systems and, in particular, complete the strange attractor bifurcation scenarios in Lorenz type systems connecting globally the homoclinic, period-doubling, Andronov–Shilnikov, and period-halving bifurcations of limit cycles.