Second Order Scalar Difference Equations

Abstract

We shall start our study with second order linear difference equations. We will show how they can be written as equivalent first order systems which have a particular form called symplectic. Later chapters will show how these symplectic systems contain discrete linear Hamiltonian systems. We will also use the linear theory in order to motivate the symplectic structure of general nonlinear discrete Hamiltonian systems. There are interconnections between these subjects and the topics of discrete variational theory, discrete matrix Riccati equations, and what we call symplectic continued fractions. But we will start with some discussion about the simplest scalar problems.