Simplicial Approximation of Poincaré Maps of Differential Equations

A method is proposed to transform a nonlinear differential system into a map without having to integrate the whole orbit as in the usual Poincaré return map technique. It consists of constructing a piecewise linear map by coarse-graining the phase surface of section into simplices and using the Poincaré return map values at the vertices to define a linear map on each simplex. The numerical results show that the simplicial map is a good approximation to the Poincaré map and it leads to a factor of 20 to 40 savings in computer time as compared with the integration of the differential equation. Computation of the generalized information dimensions of a chaotic orbit for the simplicial map gives values in close agreement with those found for the Poincaré map.