We report on a study of the forced van der Pol equation x + [epsilon](x[superscript 2] - 1)x + x = F cos[omega]t, by solving numerically the differential equation for a variety of values of the parameters [epsilon], F and [omega]. In doing so, many striking and interesting trajectories can be discovered and phenomena such as frequency entrainment,…

We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…

The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…

Results are presented of an impromptu exploration of polar formulas for volumes of revolution for certain plane regions. The material is thought to be unique, and to offer room for student exploration. It is felt pupil investigation can lead to increased pupil interest in both polar coordinates and calculus. (MP)

This note discusses the boundary in the frequency--amplitude plane for boundedness of solutions to the forced spring Duffing type equation. For fixed initial conditions and fixed parameter [epsilon] results are reported of a systematic numerical investigation on the global stability of solutions to the initial value problem as the parameters F and…