Hopf bifurcation in a system of two coupled advertising oscillators

A system of two symmetrical coupled identical oscillators, each of them being an advertising model, is considered. It possesses 4 variables and it depends on 3 parameters. The existence of five equilibrium points is found and the locus in the parameter space of values corresponding to non-hyperbolic singularities is emphasized. The topological type of the symmetric equilibrium point, which is the origin, is determined. Then the parameter values of non-degenerated or degenerated Hopf bifurcation corresponding to the origin are found by computing the first Liapunov coefficient. The center manifolds associated with these values are determined. Some 2D and 3D projections of specific phase portraits are plotted. Numerical computations in accordance with the theoretical results show the presence of stable limit cycles with 8 regimes of behavior.