Abstract

In this thesis, we discuss a new approach to the Hilbert 16th problem via computer assisted analysis. In Chapter 1, we briefly recall the basic concepts of differential equations and the history of Hilbert's 16th problem. In Chapter 2, we describe multiparameter vectors, their bifurcations and rotated vector fields. In Chapter 3, we introduce parameter continuation methods and applications to multiparameter vectors. In Chapter 4, we summarize recent studies of quadratic systems and address the most used methods, including the uniqueness theorems and classifications of Hopf bifurcations. In Chapter 5, we mention examples of cubic systems having eleven limit cycles and the cubic systems of Lienard type. In Chapter 6, we apply parameter-continuation method to compute the limit cycle bifurcation diagram for quadratic systems of special interest, whose limit cycles can not be determined with the techniques of qualitative theory. Our computations support the assumption that quadratic systems have at most four limit cycles. In Chapter 7, parameter-continuation methods are applied to investigate some Lienard cubic systems.