In this Dissertation we study the sinusoidally forced Duffing equation. We show that the Duffing equation with zero Dirichlet boundary conditions has exactly one, two or three solutions depending on the amplitude of the forcing;We also study the nonlinear reaction-diffusion equation for long time behavior using dynamical systems approach. The steady states of the reaction-diffusion equation is governed by the Duffing equation. We prove the existence of a global attractor to which all solutions approach.