A study of the periodic and quasi-periodic solutions of the discrete duffing equation

Abstract

The present investigation concentrates on the phenomenological and analytically quantitative study of the periodic and quasi-periodic solutions of a class of conservative, autonomous, nonlinear difference equations. In particular, an equation with a cubic nonlinearity, i.e., a form of the discrete Duffing equation, is studied. Following a simple analysis of the equilibrium solutions, the global structures of the phase portraits are illustrated phenomenologically for different values of the equation parameters. Three discrete perturbation procedures are then developed to obtain a consistent approximation for periodic and quasi- periodic solutions. These approximate solutions contain certain small divisors in every term other than the zero'th order term. An examination of the consequences of the vanishing of such a small divisor leads to a method of constructing exact periodic solutions in the form of finite Fourier series. The thesis concludes with a discussion of the quasi-periodic approximate solutions and their applicability.