A convergence criterion for the solutions of nonlinear difference equations and dynamical systems

Abstract

We show that the solutions of nonlinear higher order difference equations may have convergent subsequences, even when the solution as a whole does not converge, if the defining function sequence is suitably bounded near the origin. The delay size and pattern in the higher order equation play an essential role in determining which subsequences of its solutions converge. We then show that this method can be extended to planar systems and discuss applications to discrete dynamical systems that have been used in biological population models.