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Title: Homoclinic bifurcation and saddle connections for Duffing type oscillators
Author: Davenport, N. M.
ISNI:       0000 0001 3406 0042
Awarding Body: Keele University
Current Institution: Keele University
Date of Award: 1987
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The study primarily considers the nondimensionallsed Duffing's equation x" + kx' - x + x³ = F Cos ωt, (k, F, ω) > 0, and investigates the various phenomena associated with small damping and forcing amplitude. Through the use of Liapunov functions global stability of solutions is established for the forced and the unforced cases. A basic averaging process establishes a frequency/amplitude relationship for 2π/ω periodic solutions which is subsequently tested for stability of solutions: Computer plots not only reveal stable 2π/ω solutions but solutions that period double over a small ω-range when k and F are fixed bringing with them structural instability car bifurcation. Period doubling is known to be rife in one dimensional nonlinear mappings and a section is devoted to one such mapping where investigations reveal behaviour analogous to the Duffing's equation. The underlying structure of Duffing's equation is revealed through the use of the Poincaré map. The complex windings of the manifolds of the saddle points result in homoclinic intersections another type of structural instability known as homoclinic bifurcation. Before homoclinic intersection comes homoclinic tangency and this is predicted through a result obtained by Mel'nikov's method. The horseshoe map explains the complicated windings of the manifold that produce the strange attractor associated with chaotic notion. The analysis is made easier when a piecewise linear system is investigated which behaves in the same way as Duffing's equation. Coordinates of homoclinic points are found, equations of manifolds obtained and saddle connections drawn. Using a perturbed equation saddle connections of Duffing's equation are sought. The analysis unfurls simple saddle connections, double-loop, transverse and multiple loop connections. Odd periodic solutions are also investigated in a similar way. Finally, the perturbed equation is solved exactly and used to find equations of saddle connections and coordinates of homoclinic points of Duffing's equation.
Supervisor: Smith, Peter Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics