Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations

The relation between the parameters of a differential equation and corresponding discrete maps is becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. An iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODEs with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is a polynomial in the state variables and if the ODE has a fixed point at the origin. For several nonlinear systems approximations of different orders are investigated.