Abstract

This thesis concerns the asymptotic behaviour for nonlinear differential equations, and also considers how this behaviour can be recovered by appropriate numerical schemes. In particular, perturbed equations are studied, where the equation without perturbations has known asymptotic behaviour. The restoring force is generally not of linear order close to the equilibrium, and the perturbation, which is time-varying forcing function, may be very irregular. The thesis addresses three questions: first, what conditions on the forcing function characterize the case when the rate of decay of the solution of the unperturbed equation is preserved, and what is the decay rate for more slowly decaying forcing functions? Equations for which there is faster than power decay in the solution of the unperturbed equation are considered. This analysis involves generalising the class of regularly varying functions, as well generalising the notion of the Liapunov exponent to equations without leading order linear terms. Perturbation theorems, for which the decay rates of the unperturbed solutions are directly recovered, are also given. Second, we prove that continuous time behaviour can be reproduced numerically. This is done when faster-than-power law, but slower than exponential, decay occurs. A semi-implicit method is used to cope with strong global nonlinearities. If the nonlinearity is smaller than linear order close to equilibrium, a fixed step-size scheme recovers the asymptotic behaviour. Thirdly, it can be shown that the results can be applied to stochastically forced equations if the shocks have state-independent intensity. Numerical results are also presented, and the method in the deterministic case can be adapted to deal with the asymptotic behaviour of the perturbation, as well as the nonlinearity.