The Lyapunov function for the second order difference systems (Q788920)

Consider the system of difference equations (*) \(x(n+1)=ax(n)+by(n)\), \(y(n+1)=cx(n)+dy(n)\), where a,b,c,d are constant coefficients. For this system, the author constructs the following Lyapunov function \(V(x,y)=[by-(d-k)x]^ 2+[cx-(a-k)y]^ 2+(1-k^ 2)(x^ 2+y^ 2)\), \(k=a+d/\Delta +1\), \(\Delta =ad-bc\). When the inequalities \(1-\Delta^ 2>0\), \(1-k^ 2>0\) are satisfied, the function V(x,y) is positive definite and its first difference \(\Delta\) V(x,y) is negative definite. The function V is used for obtaining estimates of the attraction region of the nonlinear difference system for which (*) is the system of first approximation: \(x(n+1)=ax(n)+by(n)+\gamma x(n)^ 3\), \(y(n+1)=cx(n)+dy(n)+\beta y(n)^ 3\), where \(\beta\), \(\gamma\) are constants and \(| \beta |<| \gamma |\). The function V is also used for analyzing the stability of the second order difference system with variable coefficients \(x(n+2)+2Ax(n+1)+Bx(n)=0\), where A and B are constant coefficients.