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UBC Theses and Dissertations

Melnikov's method with applications Chow, Yan Kin

Abstract

This thesis gives a detailed discussion of Melnikov's method, which is an analytical tool to study global bifurcations that occur in homoclinic or heteroclinic loops, or in one-parameter families of periodic orbits of a perturbed system. Basic results of the Melnikov theory relating the number, positions and multiplicities of the limit cycles by the number, positions and multiplicities of the zeros of the Melnikov function are proved. We then give several examples to illustrate the theory. In particular, we use the Melnikov theory to study the exact number of limit cycles in the Bogdanov-Takens system with reflection symmetry. We then extend the first-order Melnikov theory to higher-order and establish some results relating the number, positions and multiplicities of the limit cycles by the number, positions and multiplicities of the zeros of the first non-vanishing Melnikov function. Next, we derive a formula for the second-order Melnikov function for certain perturbed Hamiltonian systems using Franchise's recursive algorithm. Finally, this formula is applied to an example.

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