Summary
The Hopf calculations allow us to estimate the amplitude of the limit cycle emerging close to the bifurcation point analytically. According to the Hopf theorem, the radius of the limit cycle grows as a square root function of the bifurcation parameter. Many mechanical models consider exactly third degree nonlinearities in the Taylor expansion, see for instance the Duffing oscillator. However, second degree terms break the symmetry of the nonlinearities and that of the emerging limit cycle branches. This way, the usual analytic prediction of the limit cycles might be inaccurate for these asymmetric cases, that is, an estimation with the calculated radius is not satisfactory. This work presents the possibilities of correction, and some oscillatory systems where these corrections are relevant and necessary.
