Keele Research Repository

The study primarily considers the nondlmensionallsed Buffing's equation
x" + kx' - x + x3 = F Cos wt, (k, F, ?) > 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 2«/u periodic solutions which is subsequently tested for stability of solutions: Computer plots not only reveal stable 2ir/u solutions but solutions that period double over a small wränge when k and F are fixed bringing with them structural instability car bifurcation. Period doubling is known to be rite in one dimensional nonlinear mappings and a section is devoted to one such mapping where investigations reveal behaviour analogous to the Buffing's equation.
The underlying structure of Buffing's equation is revealed through the use of the Poincar^ 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 riel'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 Ouffing'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 buffing's equation.