Periodic convergence in the discrete hungry Toda equation

Abstract

In this paper, we clarify the periodic behaviour in the discrete hungry Toda (dhToda) equation, which is an extension of the famous integrable discrete Toda equation. The centre manifold theory, which is a classical analysis theory, plays a key role in analysing the dhToda equation. We first observe the asymptotic behaviour of the dhToda variable through a numerical example. We then discuss the existence of a centre manifold associated with the dhToda equation. Finally, we find this centre manifold and then analyse the asymptotic behaviour of the dhToda variable as the discrete-time variable goes to infinity.