POLYNOMIAL-APPROXIMATION OF POINCARE MAPS FOR HAMILTONIAN-SYSTEMS

Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a hamiltonian one, into a map without having to integrate the whole orbit as in the well known P oincare map technique. We construct piecewise polynomial maps by coars e-graining the phase surface of section into parallelograms using valu es of the Poincare maps at the vertices to define a polynomial approxi mation within each cell. The numerical experiments are in good agreeme nt with the standard map taken as a model problem. The agreement is be tter when the number of vertices and the order of the polynomial fit i ncrease. The synthetic mapping obtained is not symplectic even if at v ertices there is an exact. interpolation. We introduce a second new me thod based on a global fitting. The polynomials are obtained using at once all the vertices and fitting by least square polynomes but in suc h a way that the symplectic character is not lost.