Nonlinear Phenomena in Complex Systems, 2002, Vol.5, No.2, pp.137-150

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2002, Vol.5, No.2, pp.137-150

On the Andronov-Hopf Bifurcation. pp.137-150
V. A. Gaiko

Two-dimensional quadratic dynamical systems are mainly considered. We study the Andronov--Hopf bifurcation by means of canonical systems with field-rotation parameters. Applying such systems, we construct, for example, a quadratic system with at least four limit cycles in $(3:1)$ distribution and develop techniques of the functions of limit cycles for the investigation of various limit cycle bifurcations. All these results will be used further for the study of local bifurcation surfaces and global families of multiple limit cycles, and will be applied to the solution of Hilbert's Sixteenth Problem on the maximum number and relative position of limit cycles of arbitrary polynomial systems.
Key words: Hilbert's Sixteenth Problem, Andronov-Hopf bifurcation, field-rotation parameter, limit cycle

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