Non‐linear differential equations and the method of perturbation

Abstract

The limit‐cycle solution of van der Pol's non‐linear differential equation is found approximately by the method of perturbation; this is suitable when the non‐linearity is weak. At each stage of the perturbation a linear differential equation is solved and two unknown constants of integration arise. The text‐books usually determine these constants by specifying zero initial conditions for the perturbation and its derivative. This procedure cannot be justified when finding a limit‐cycle of a non‐linear differential equation and it leads to some wrong values. The method used here is to choose the constants of integration so as to minimize the residual error when the truncated solution is substituted in the differential equation. It is then found that each perturbation depends upon the position of truncation. Numerical values are given for truncation at the first and second perturbations.