On the geometry of polynomial dynamical systems

In this paper, the author performs a global qualitative study of plane polynomial dynamical systems and suggests a new geometric approach to solving the sixteenth Hilbert problem on the maximum number and mutual location of their limit cycles in two special cases of such systems. First of all, using the geometric properties of four parameters rotating the vector field of a new canonical system constructed in the paper, the author proposes the proof of his early conjecture, which asserts that the maximum number of limit cycles of an arbitrary quadratic system is equal to 4, and their location (3: 1) is uniquely possible 4]. Then using the same geometric approach, the author solves the primary problem for the polynomial Liénard system (in this special case, it is considered as the thirteenth Smale problem), and generalizing the obtained results, the author formulates the theorem on the maximum number of limit cycles enclosing one singular point in the case of a polynomial system. Moreover, applying the Wintner-Perko termination principle for multiple limit cycles, the author develops an alternative approach to solving the sixteenth Hilbert problem, and using this approach, the author completes the global qualitative study of a general cubic Liénard system having three singular points in the finite part of the plane. In conclusion, the author discusses one more known approach to solving the sixteenth Hilbert problem. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.