A Family of Periodic Functions
by
Harley Flanders
University of north Florida

Consider the system

dx/dt=-y2j-1    dy/dt=x2k-1

where j, k are positive integers, and we impose the initial conditions x(0)=1, y(0)=0. It turns out that the solution is periodic, with half-period p_{i,  k}. Using our software package ODE, which is based on automatic differentiation, we have computed these p_{i,  k} for the indices up to 63. An extensive analysis of these numbers suggested relations between them, and asymptotic behavior. We have proved some of these observations.

It turns out the the trajectories are closed convex curves bounding areas closely related to the half-periods. We have computed these areas by a numerical integration technique, also based on automatic differentiation, in order to check our previous computations.

http://homepages.feis.herts.ac.uk/~comqun/AD2000/Ext_Abstracts/flanders1.ps

Date received: February 10, 2000