Machine-assisted discovery of integrable symplectic mappings

Abstract

I. Introduction II. 2D symplectic mappings and integrability: historical overview III. 1-piece maps A. Polygon maps with integer coefficients B. Fundamental polygons of the first kind IV. 2-piece maps A. Linear mappings 1. Polygon maps with integer coefficients 2. Fundamental polygons of the second kind 3. Ingredients of mappings with polygon invariant B. Nonlinear mappings 1. Zoo mappings 2. Nonlinear integrable maps with polygon invariants V. Search algorithm VI. 3-piece maps A. Isolated d B. Discrete d, (D) C. Continuous d, (C) VII. 4-piece maps A. Isolated d and r B. Isolated d, discrete r C. Isolated d, continuous r D. Additional cases 1. Continuous and semi-continuous d with unity r. 2. Non-unity values of r. VIII. Applications of mappings with polygon invariants A. Connection to Painlevé equations and Suris mappings B. Near-integrable systems C. Discrete perturbation theory IX. Summary X. Acknowledgments A. Calculation of action-angle variables for polygon mappings 1. Action 2. Rotation number and angle variable a. Method I b. Method II B. Pseudo-code for the search algorithm C. Action-angle variables for 4-piece mappings D. Properties of invariant polygons for Zoo maps References