This textbook contains the lecture series originally delivered at the "Advanced Course on Limit Cycles of Differential Equations" in the Centre de Recerca Matemàtica Barcelona in 2006.
The topics covered are the center-focus problem for polynomial vector fields, and the application of Abelian integrals to limit cycle bifurcations. Both topics are related to Hilbert's sixteenth problem. In particular, the book will be of interest to students and researchers working in the qualitative theory of dynamical systems.
This second edition provides updates, further clarifications and remarks, and includes an expanded list of references.
Part I: Around the Center-Focus Problem.- 1 Centers and Limit Cycles.- 2
Darboux Integrability.- 3 Liouvillian Integrability.- 4 Symmetry.- 5 Cherkas
Systems.- 6 Monodromy.- 7 The Tangential Center-Focus Problem.- 8 Monodromy
of Hyperelliptic Abelian Integrals.- 9 Holonomy and the Lotka-Volterra
System.- 10 Other Approaches.- Part II: Abelian Integrals and Applications to
the Weak Hilberts 16th Problem.- 1 Hilberts 16th Problem and Its Weak
Form.- 2 Abelian Integrals and Limit Cycles.- 3 Estimate of the Number of
Zeros of Abelian Integrals.- 4 A Unified Proof of the Weak Hilberts 16th
Problem for n=2.
Colin Christopher is Associate Head of School at the University of Plymouth in Devon, UK.Chengzhi Li is Professor Emeritus at the Peking University in China.
Joan Torregrosa is Associate Professor at the Universitat Autònoma de Barcelona in Bellaterra, Spain.
