Generalized fourier series for the study of limit cycles

The approximate solution, to first order, of non-linear differential equations is studied using the method of harmonic balance with generalized Fourier series and Jacobian elliptic functions. As an interesting use of the series, very good analytic approximations to the limit cycles of Liénard's ordinary differential equation (ODE), X ̈ + g(X) = f(X) X ̇ , are presented. Specifically, it is shown that, contrary to an opinion given in a well-known textbook on non-linear oscillations, g( X) not only modifies the period but influences the topology. In the generalized van der Pol equation with f( X) = ε(1− X 2) and g( X) = AX + 2 BX 3 for ϵ < 0·1, the presence of zero, one, or three limit cycles is found to depend on the value of A B .