Global bifurcations and chaos in polynomial dynamical systems

Polynomial dynamical systems are considered. First of all, we study global bifurcations of multiple limit cycles in two-dimensional systems developing a new approach to Hilbert\´s Sixteenth Problem on the maximum number and relative position of limit cycles. The problem has not been solved completely even for the simplest nonlinear case: for the case of quadratic systems, and we suggest a program solving this problem in the quadratic case. This approach can be applied also to the study of arbitrary polynomial systems and to the global qualitative analysis of higher-dimensional dynamical systems. In particular, we discuss how to apply the obtained results for the construction of a three-dimensional system with a "strange attractor" on the base of a planar quadratic system with two unstable foci and an invariant straight line. This study could give us a chaos birth bifurcation in the polynomial dynamical systems.