Superharmonic solutions of nonlinear differential equations

Abstract

This thesis is a study of the structure of superhaxmonic solutions of order m, to the sloshing equation introduced by Chester and Ockendon &; Ockendon, and to a lesser extent, Duffing’s equation. We use the Lyapunov Schmidt procedure to reduce these problems to two bifurcation equations. We elucidate the form and leading terms of the bifurcation equations. The usual scaling techniques fail when superharmonics of order 4 or greater are sought. An alternative scaling method is provided, which works for superharmonic solutions of all orders. The method is rigorous, and naturally provides an explicit approximation to the bifurcation surface. To produce a formula for the approximate bifurcation surface it is necessary to explicitly calculate coefficients in the bifurcation equations. A simple algorithm, which calculates the terms which may be required, is given. The method is implemented using Macsyma. The program, TAYLOR, produces the information for superharmonic and subharmonic solutions for a large class of nonlinear oscillation problems