Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation

This paper concerns the quadratically-damped Mathieu equation: x ̈ +(δ+ε cos t)x+ x ̇ | x ̇ |=0. Numerical integration shows the existence of a secondary bifurcation in which a pair of limit cycles come together and disappear (a saddle-node of limit cycles). In δ– ε parameter space, this secondary bifurcation appears as a curve which emanates from one of the transition curves of the linear Mathieu equation for ε≈1.5. The bifurcation point along with an approximation for the bifurcation curve is obtained by a perturbation method which uses Mathieu functions rather than the usual sines and cosines.