First Liapunov coefficient for coupled identical oscillators. Application to coupled demand-supply model

A general formula for the computation of the first Liapunov coefficient corresponding to the Hopf bifurcation in a four-dimensional system of two coupled identical oscillators is performed for two cases. Only bi-dimensional vectors are involved. Then a model of two coupled demand-supply systems, depending on four parameters is considered. A study of the Hopf bifurcation is done around one of the symmetrical equilibrium, as the parameters vary. The loci in the parameter space of the parameters values corresponding to subcritical, supercritical or degenerated Hopf bifurcation are found. The computation of the Liapunov coefficients is done using the derived formula. Numerical plots emphasizing the existence of different types of limit cycles are developed.