دانلود کتاب چرخه‌های حدی و شبکه‌های هموکلینیک در سیستم‌های چندجمله‌ای دوبعدی

This book is a monograph about limit cycles and homoclinic networks in polynomial systems. The study of dynamical behaviors of polynomial dynamical systems has been stimulated by the Hilbert’s 16th problem in 1900. Too many scientists have tried to work on the Hilbert sixteenth problem. Until now, no significant results have been achieved yet.
In this book, the properties of equilibriums in planar polynomial dynamical systems are studied. The homoclinic networks of sources, sinks, and saddles in self-univariate polynomial systems are discussed, and the corresponding bifurca-tion theory is developed. The corresponding first integral manifolds are determined analytically, and networks of source, sinks, and saddles are illustrated. The homo-clinic networks of saddles and centers (or limit cycles) in crossing-univariate polyno-mial systems are discussed, and the corresponding bifurcation theory is developed. The corresponding first integral manifolds are polynomial functions. The homoclinic networks of saddles and centers are illustrated, which are without any sources and sinks. Since the maximum numbers of equilibriums for such two types of planar polynomial systems with the same degrees are discussed, the maximum centers and saddles in homoclinic networks are obtained, and the maximum numbers of sinks, sources, and saddles in homoclinic networks without centers are obtained as well. Such studies are to achieve global dynamics of planar polynomial dynamical systems, which can help one study global behaviors in nonlinear dynamical systems in physics, chemical reaction dynamics, engineering dynamics, and so on.
In this book, five chapters are included. Chapter 1 is for instruction to polynomial dynamical systems. Chapter 2 discusses homoclinic networks without centers. The corresponding bifurcations for homoclinic networks without centers are presented in Chap. 3. Chapter 4 discusses homoclinic networks with centers and saddles. The corresponding bifurcations for homoclinic with centers are presented in Chap. 5.
Finally, the author hopes the materials presented herein can provide a better understanding of nonlinear dynamics in science and engineering.