Consider the differential system (*) \(dx/dt =f(t,x)\) having \(x=0\) as unique stationary solution. Assume that there exists a decomposition of (*) in the form \[ dx_1/dt =f_1(t, x_1) +h_1(t,x_1, x_2),\;dx_2/dt= f_2(t,x_2) +h_2(t,x_1,x_2). \tag{**} \] Furthermore, assume there exists a decomposition of the systems \(dx_k/dt= f_k(t,x_k)\), \(k=1,2\), into the subsystems \[ dx_{1i}/dt= f_{1i} (t,x_{1i}) +h_{1i} (t,x_1),\;dx_{2i}/dt =f_{2i} (t,x_{2i}) +h_{2i} (t,x_2),\;i=1,2. \tag{***} \] The key idea to construct a hierarchical Lyapunov function for (*) is to construct a matrix function \[ U(t,x)= \left(\begin{matrix} U_1(t,x_1) & U_3(t,x) \\ U_3(t,x) & U_2(t,x_2) \end{matrix} \right) \] where \(U_1\) and \(U_2\) are constructed for the subsystems (***) while \(U_3\) is determined by the interconnection functions \(h_i\) in (**).