Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.1, pp.11--27

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2000, Volume 3, Number 1, pp.11--27

On Global Bifurcations and Hilbert's Sixteenth Problem.
Valery A. Gaiko

Two-dimensional polynomial dynamical systems are mainly considered. We develop Erugin's two-isocline method for the global analysis of such systems, construct canonical systems with field-rotation parameters and study various limit cycle bifurcations. In particular, we show how to carry out the classification of separatrix cycles and consider the most complicated limit cycle bifurcation: the bifurcation of multiple limit cycles. Using the canonical systems, cyclicity results and Perko's termination principle, we outline a global approach to the solution of Hilbert's 16th Problem. We discuss also how to generalize this approach for the study of higher-dimensional dynamical systems and how to apply it for systems with more complicated dynamics.
Key words: Hilbert's 16th Problem, Erugin's two-isocline method, Wintner's principle of natural termination, Perko's planar termination principle, field-rotation parameter, bifurcation, limit cycle, separatrix cycle.

Full text: Acrobat PDF (195KB) PostScript (905KB) PostScript.gz (196KB)