Numerical study of quadratic area-preserving mappings

Abstract

Dynamical systems with two degrees of freedom can be reduced to the study of an area-preserving mapping. We consider here, as a model problem, the mapping given by the quadratic equations: x 1 = x cos ⁡ α − ( y − x 2 ) sin ⁡ α {x_1} = x\cos \alpha - \left ( {y - {x^2}} \right )\sin \alpha , y 1 = x sin ⁡ α + ( y − x 2 ) cos ⁡ α {y_1} = x\sin \alpha + \left ( {y - {x^2}} \right ) \\ \cos \alpha , which is shown to be in a sense the simplest nontrivial mapping. Some analytical properties are given, and numerical results are exhibited in Figs. 2 to 14.