
PresentationsProblems in theory of lowdimensional dynamical systemsUnited Institute of Informatics Problems, National Academy of Sciences of Belarus, Leonid Beda Str. 64, Minsk 220040, Belarus We discuss some problems of the global qualitative analysis of lowdimensional polynomial dynamical systems. To control all of the limit cycle bifurcations in such systems, especially, bifurcations of multiple limit cycles, 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 bifurcational geometric methods based on the wellknown Weierstrass preparation theorem and the Perko planar termination principle stating that the maximal oneparameter family of multiple limit cyclesterminates either at a singular point which is typically of the same multiplicity (cyclicity) or on a separatrix cycle which is also typically of the same multiplicity (cyclicity). This principle is a consequence of the principle of natural termination which was stated for higherdimensional dynamical systems by A.\,Wintner who studied oneparameter families of periodic orbits of the restricted threebody problem and used Puiseux series to show that in the analytic case any oneparameter family of periodic orbits can be uniquely continued through any bifurcation except a perioddoubling bifurcation. Such a bifurcation can happen, e.\,g., in a threedimensional Lorenz system. But this cannot happen for planar systems. That is why the WintnerPerko termination principle is applied for studying multiple limit cycle bifurcations of planar polynomial dynamical systems. If we do not know the cyclicity of the termination points, then, applying canonical systems with field rotation parameters, we use geometric properties of the spirals filling the interior and exterior domains of limit cycles. Applying this method, we have solved, e.\,g., Smale's Thirteenth Problem for the classical Li\'{e}nard system. Generalizing the obtained results, we have solved Hilbert's Sixteenth Problem on the maximum number and distribution of limit cycles for the Kukles cubiclinear system and for the general Li\'{e}nard polynomial system with an arbitrary number of singular points. We discuss also how to apply this approach for studying global limit cycle bifurcations of discrete and continuous Hollingtype systems which model the population dynamics in biomedical and ecological systems. Finally, applying a similar approach, we consider various applications of threedimensional polynomial dynamical systems and, in particular, complete the strange attractor bifurcation scenario in the classical Lorenz system globally connecting the homoclinic, perioddoubling, AndronovShilnikov, and periodhalving bifurcations of its limit cycles.
