Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Pei Yu

Abstract

This thesis contains two parts. In the first part, we investigate bifurcation of limit cycles around a singular point in planar cubic systems and quadratic switching systems. For planar cubic systems, we study cubic perturbations of a quadratic Hamiltonian system and obtain 10 small-amplitude limit cycles bifurcating from an elementary center, for which up to 5th-order Melnikov functions are used. Moreover, we prove the existence of 12 small-amplitude limit cycles around a singular point in a cubic system by computing focus values. For quadratic switching system, we develop a recursive algorithm for computing Lyapunov constants. With this efficient algorithm, we obtain a complete classification of the center conditions for a switching Bautin system. Moreover, we construct a concrete example of switching system to obtain 10 small-amplitude limit cycles bifurcating from a center.

In the second part, we derive two explicit, computationally explicit, recursive formulas for computing the normal forms, center manifolds and nonlinear transformations for general n-dimensional systems, associated with Hopf and semisimple singularities, respectively. Based on the formulas, we develop Maple programs, which are very convenient for an end-user who only needs to prepare an input file and then execute the program to “automatically” generate the results. Several examples are presented to demonstrate the computational efficiency of the algorithms. In addition, we show that a simple 3-dimensional quadratic vector field can have 7 small-amplitude limit cycles, bifurcating from a Hopf singular point. This result is surprising higher than the Bautin’s result for quadratic planar vector fields which can only have 3 small-amplitude limit cycles around an elementary focus or an elementary center.

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