A classification of the rational integrable generalized standard maps

Abstract

This dissertation examines the integrability of certain planar maps. An integral of a planar map is a real-valued function that remains invariant under composition with the map. In particular we classify the generalized standard maps, maps of the form φ(x, y) = (f(x)-y, x), that have a polynomial integral and those that have a rational integral. Chapter 1 brings to light the pertinent definitions. Here we show numerical examples of integrability, non-integrability and questionable integrability. Chapter 1 concludes with a discourse on linear maps, giving a classification of those with a polynomial or rational integral. The meat of this work lies in Chapter 2, which contains the classification of the generalized standard maps. Here we give a short list of the maps that have a polynomial integral as well as an integral for each case. Finally we show that if a generalized standard map has a rational integral then it has a polynomial integral. Deciding if a planar map is integrable or deciding what family the integral lies in is one of the difficult tasks assumed by the field of dynamical systems. Here we have approached the integrability question from an algebraic viewpoint. Some attempt has been made to make the proofs simple yet complete, without using the heavy-duty analytical tools that can often be found in the analysis of a dynamical systems.