This book contains the lecture series delivered at the "Advanced Course on Limit Cycles of Differential Equations." It covers the center-focus problem for polynomial vector fields and the application of abelian integrals to limit cycle bifurcations.

This book contains the lecture series originally delivered at the Advanced Course on Limit Cycles of Differential Equations in the Centre de Rechercha Mathematica Barcelona in 2006. It covers the center-focus problem for polynomial vector fields, and the application of abelian integrals to limit cycle bifurcations. Both topics are related to the authors' interests in Hilbert's sixteenth problem, but would also be of interest to those working more generally in the qualitative theory of dynamical systems.

Gives a survey of current areas of active research

Written in an approachable manner suitable for graduate students

Includes supplementary material: sn.pub/extras

Inhalt
Around the Center-Focus Problem.- Centers and Limit Cycles.- Darboux Integrability.- Liouvillian Integrability.- Symmetry.- Cherkas' Systems.- Monodromy.- The Tangential Center-Focus Problem.- Monodromy of Hyperelliptic Abelian Integrals.- Holonomy and the Lotka-Volterra System.- Other Approaches.- Abelian Integrals and Applications to the Weak Hilbert's 16th Problem.- Hilbert's 16th Problem and Its Weak Form.- Abelian Integrals and Limit Cycles.- Estimate of the Number of Zeros of Abelian Integrals.- A Unified Proof of the Weak Hilbert's 16th Problem for n=2.