An elementary approach to a generalized Hopf bifurcation
For a system of autonomous differential equations undergoing a Hopf bifurcation the bifurcation equation is derived. The zeros of the bifurcation equation correspond to periodic solutions of the differential equations. The reduction is performed by the Lyapunov-Schmidt method and by expanding the bifurcation equation into Fourier series. The coefficients are obtained by integrating a recursive set of differential equations, which is stated only symbolically. Finally the well known stability condition for the periodic solution in terms of the derivative of the bifurcation equation is stated and the procedure is compared shortly with the calculations using Center manifolds.
system of autonomous differential equations
scientific article; zbMATH DE number 488211