Approximate first-order solutions of non-linear dynamical mappings

We present a simple method to construct approximate analytical solutions of a class of non-linear dynamical mappings in Rd(d=1,2,…). This class is defined as a system of d non-linear first-order difference equations. This system is a discrete analogue of a system of d-first-order ordinary differential equations which appear in many important problems of mechanics, physics and engineering. We apply our method to derive approximate analytical solutions of some model maps in R and R2. The analytical results are tested numerically with the iterates of the maps and good agreement is found for reasonably large values of the non-linearity. The analytical results obtained for the well-known Hénon map are identical to the results obtained with normal forms at the lowest order, up to a remainder of higher order. The coefficients of a continuous Fourier interpolation of the orbits are also explicitly obtained. The advantage of the proposed method lies in its simplicity (compared with other methods) and ease of application. The approximate solutions of these maps can be used as a theoretical guidance for further numerical or analytical studies, e.g. stability analysis and control of chaos.