The problem consists of a second order differential equation z"=f(z,z'), rewritten to first order form, thus providing a system of ordinary differential equations of dimension 2. It was proposed by B. van der Pol in the 1920's and describes the behaviour of nonlinear vacuum tube circuits.
It has two periodic solutions, the constant solution, z(t)= 0, that is unstable, and a nontrivial periodic solution that correspond to an attractive limit cycle. The equation depends on a parameter that weights the importance of the nonlinear part of the equation.
In the tests two different versions of the equation are considered: z"=f1(z,z')=µ(1-z²)z'-z
and u"=f2(u,w)=((1-u²)w-u)/eps, with eps=1/µ².
The latter form, obtained by means of a scaling transformation of the former, has been introduced to make the steady-state
approximation independent of the parameter. The parameters have been fixed to µ=1e3 and eps=1e-6, respectively, giving rice to a stiff problem.
A complete description of the problem could be found in
vdpol.pdf
