Stability criteria for certain classes of nonlinear difference equations

Abstract

The general mth order difference equation X( n+ m)+ a 1 X( n+ m−1)+…+ a m X( n) = F[ n, X( n),…, X( n+ m−1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V( n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV( n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented. The results of the present approach in the homogeneous case where F[ n, X( n),…, X( n+ m−1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good.