Mathematical analysis of the smallest chemical reaction system with Hopf bifurcation (Q1360642)

Consider the system \[ dx/dt= kx- k_2xy,\quad dy/dt=- k_3y+ k_5z,\quad dz/dt= k_4- k_5z\tag{\(*\)} \] in the positive orthant with given \(k_i>0\) and \(k\) as bifurcation parameter. \((*)\) has two equilibria, the trivial steady state \(\overline x=\overline y=\overline z=0\) and a nontrivial one \((x_s(k),y_s(k),z_s(k))\). For \(k=0\) both steady states coincide. The author proves that this multiple equilibrium point is globally stable in the positive orthant. Moreover, the existence of a critical value \(k_c\) is established connected with the bifurcation of a stable limit cycle from \((x_s(k_s),y_s(k_s),z_s(k_s))\) as \(k\) increases. Finally, a Lyapunov function is derived characterizing a region \(G_s\) such that all trajectories of \((*)\) starting in \(G_s\) tend to \((x(k)_s,y(k)_s,z(k)_s)\) as \(t\to\infty\) and for \(0<k<k_s\).