Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models (Q1019222)

The Hopf bifurcation in the following 4D system is studied \[ \begin{aligned}\dot{x_1} &= -a(x_1+bx_2+2x_1x_2+x_2^2+x_1x_2^2)+c_1(x_1-x_3))\\ \dot{x_2} &= x_1+x_2+2x_1x_2+x_2^2+x_1 x_2^2 \\ \dot{x_3} &= -a(x_3+bx_4+2x_3 x_4 +x_4^2 +x_3 x_4^2) +c_2(x_3 - x_1)\\ \dot{x_4} &= x_3 +x_4 +2x_3 x_4 +x_4^2 +x_3 x_4^2.\end{aligned} \] Formulas for the computation of the first Lyapunov coefficient are obtained in two cases of Hopf bifurcation. A study of the Hopf bifurcation around the symmetric equilibrium point is performed using the general formulas obtained for the computation of the Lyapunov coefficients. The generalized Hopf bifurcation (Bautin) is illustrated by extensive numerical computation. The results obtained when the oscillators are non-symmetrically coupled are compared with those when they are symmetrically coupled. From economical point of view, the objective of these studies is to determine the parameters for which the future behavior of the variable is table or periodic. In terms of the bifurcation theory this means determination of regions of the parameters space for which attractors or limit cycles with attractive properties exist. It is to be noted that studied 4D system is a generalization of the classic (in economic dynamics) dynamic system \[ \begin{aligned} \frac{dx}{d\tau} &= k-\alpha x y^2 +\beta y\\ \frac{dy}{d \tau} &= \alpha x y^2 - \delta y.\end{aligned} \]