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Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Yu, Pei

Abstract

Oscillation is a common natural phenomenon in real world problems. The most efficient mathematical models to describe these cyclic phenomena are based on dynamical systems. Exploring the periodic solutions is an important task in theoretical and practical studies of dynamical systems.

Abelian integral is an integral of a polynomial differential 1-form over the real ovals of a polynomial Hamiltonian, which is a basic tool in complex algebraic geometry. In dynamical system theory, it is generalized to be a continuous function as a tool to study the periodic solutions in planar dynamical systems. The zeros of Abelian integral and their distributions provide the number of limit cycles and their locations.

In this thesis, we apply the Abelian integral method to study the limit cycles bifurcating from the periodic annuli for some hyperelliptic Hamiltonian systems. For two kinds of quartic hyperelliptic Hamiltonian systems, the periodic annulus is bounded by either a homoclinic loop connecting a nilpotent saddle, or a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. For a quintic hyperelliptic Hamiltonian system, the periodic annulus is bounded by a more degenerate heteroclinic loop, which connects a nilpotent saddle to a hyperbolic saddle. We bound the number of zeros of the three associated Abelian integrals constructed on the periodic structure by employing the combination technique developed in this thesis and Chebyshev criteria. The exact bound for each system is obtained, which is three. Our results give answers to the open questions whether the sharp bound is three or four. We also study a quintic hyperelliptic Hamiltonian system with two periodic annuli bounded by a double homoclinic loop to a hyperbolic saddle, one of the periodic annuli surrounds a nilpotent center. On this type periodic annulus, the exact number of limit cycles via Poincar{\'e} bifurcation, which is one, is obtained by analyzing the monotonicity of the related Abelian integral ratios with the help of techniques in polynomial boundary theory. Our results give positive answers to the conjecture in a previous work.

We also extend the methods of Abelian integrals to study the traveling waves in two weakly dissipative partial differential equations, which are a perturbed, generalized BBM equation and a cubic-quintic nonlinear, dissipative Schr\"{o}dinger equation. The dissipative PDEs are reduced to singularly perturbed ODE systems. On the associated critical manifold, the Abelian integrals are constructed globally on the periodic structure of the related Hamiltonians. The existence of solitary, kink and periodic waves and their coexistence are established by tracking the vanishment of the Abelian integrals along the homoclinic loop, heteroclinic loop and periodic orbits. Our method is novel and easily applied to solve real problems compared to the variational analysis.

Summary for Lay Audience

Periodic motions appear in almost all natural and engineering dynamical systems. Determining the number of periodic solutions and their locations plays an important role in solving real world problems, in particular on stability and bifurcations of the system. It is important to determine what may cause oscillation and what may destroy oscillation, and what affects the period and amplitude of oscillation. However, it is not easy to determine all possible locations, periods and amplitudes even for the oscillations in two-dimensional dynamical systems. In this thesis, we apply an integral defined on a continuous ovals as a bifurcation function to study the oscillation phenomena in some perturbed temporal or temporal-spatial dynamical systems, and obtain new results which solve open problems in the existing literature.

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