Numerical study of quadratic area-preserving mappings

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: <!-- MATH ${x_1} = x\cos \alpha - \left( {y - {x^2}} \right)\sin \alpha$ --> , <!-- MATH ${y_1} = x\sin \alpha + \left( {y - {x^2}} \right) \\\cos \alpha$ --> , 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.