Quadratic Forms as Lyapunov Functions in the Study of Stability of Solutions to Difference Equations

A system of linear autonomous difference equations x(n + 1) = Ax(n) is considered, where x ∈ Rk, A is a real nonsingular k × k matrix. In this paper it has been proved that if W (x) is any quadratic form and m is any positive integer, then there exists a unique quadratic form V (x) such that ∆mV = V (Amx) − V (x) = W (x) holds if and only if μiμj 6= 1 (i = 1, 2 . . . k; j = 1, 2 . . . k) where μ1, μ2, . . . , μk are the roots of the equation det(Am − μI) = 0. A number of theorems on the stability of difference systems have also been proved. Applying these theorems, the stability problem of the zero solution of the nonlinear system x(n + 1) = Ax(n) + X(x(n)) has been solved in the critical case when one eigenvalue of a matrix A is equal to minus one, and others lie inside the unit disk of the complex plane.