Normally we think of a pendulum as a weight suspended by a flexible string or cable, so that it may swing back and forth. Another type of pendulum consists of a weight attached by a light (but inflexible) rod to an axle, so that it can swing through larger angles, even making a 360 rotation if given enough velocity.

Though it is not precisely correct in practice, we often assume that the magnitude of the frictional forces that eventually slow the pendulum to a halt is proportional to the velocity of the pendulum. Assume also that the length of the pendulum is 1 meter, the weight at the end of the pendulum has mass 1 kg, and the coefficient of friction is 0.5. In that case, the equations of motion for the pendulum are

where t represents time in seconds, x represents the angle of the pendulum from the vertical in radians (so that x = 0 is the rest position), y represents the velocity of the pendulum in radians per second, and 9.81 is approximately the acceleration due to gravity in meters per second squared. Here is a phase portrait of the solution with initial position x(0) = 0 and initial velocity y(0) = 5. This is a graph of x versus y as a function of t, on the time interval 0 ? t ? 20.

<b class="bold">g = inline('[x(2); -0.5*x(2) - 9.81*sin(x(1))]', 't', 'x');[t,...