The Solutions of a Nonlinear Difference Equation Found in Numerical Analysis

The solutions of a nonlinear difference equation resulting from the trapezoidal quadrature approximation of a piecewise linear differential equation are examined. A phase plane mapping technique is used to find the periodic and unbounded solutions of the difference equation and to determine the stability of these solutions. From the phase plane mappings of the unbounded solutions it is shown that the energy of the bounded solutions can grow and decay by large amounts. The maximum rate of Hamiltonian increase of the unbounded solutions is calculated from the mapping transformations. It is shown that the stability of a solution can only be guaranteed for discrete ranges of the time step with fixed initial conditions. The solutions of the non-autonomous difference equations for a sinusoidal forcing term and the damped difference equations are also examined.