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- Melnikov's method with applications
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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.
Item Metadata
Title |
Melnikov's method with applications
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2001
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Description |
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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Extent |
2938276 bytes
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Genre | |
Type | |
File Format |
application/pdf
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Language |
eng
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Date Available |
2009-07-30
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0079999
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2001-05
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.