Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models

Two identical 2D dynamical systems written in a general form and depending on a parameter were coupled linearly and non-symmetrically. The Hopf bifurcation in the 4D system is studied. New formulae for the computation of the first Lyapunov coefficient are obtained in two cases of Hopf bifurcation. They use only 2D vectors and generalize the expressions of the first Lyapunov coefficient deduced in [13] for symmetrically coupled dynamical systems. In addition, formulae for the computation of the second Lyapunov coefficient are obtained in terms of 2D vectors.
A particular 4D dynamical system obtained by coupling non-symmetrically two identical 2D advertising models is considered. It depends on 4 parameters, two of them being the coupling parameters. A study of the Hopf bifurcation around the symmetric equilibrium point is performed using the general formulae obtained by us for the computation of the Lyapunov coefficients. Even if the Hopf bifurcation in the single system is always supercritical, for the coupled system the Hopf bifurcation can be supercritical, subcritical or degenerate. The generalized Hopf bifurcation (Bautin) is illustrated by our numerical computations using the software winpp [4]. The results obtained when the oscillators are non-symmetrically coupled are compared with those when they are symmetrically coupled.