A Straightforward Approach To Solve Ordinary Nonlinear Differential Systems

Abstract

The analytical solution of nonlinear differential systems is addressed. The
approach consists in algebraic series in the time variable that leads to elementary recurrence
algorithms. This is an alternative to standard techniques for numerical integration and it
ensures the theoretical exactness of the response. Here it is shown that the systematic
extension to nonlinear problems leads to a very convenient numerical integration algorithm
without any approximation or truncation in each time step as is the case with the standard
integration schemes (Newmark, α -method, Wilson-θ , Runge-Kutta).
Most of usual nonlinearities may be represented by an algebraic series. The calculation of the
derivatives is immediate. Factors of the different powers are equated giving rise to the
algebraic recurrence algorithm. Since a unitary domain is convenient, a
nondimensionalization of the variable t is done using T, an interval of interest τ = t T .
When problems of numerical divergence appear, steps of appropriate duration T are used.
Several examples complete this study. They are: a) projectile motion, b) N bodies with
gravitational attraction, c) Lorenz equations, d) Duffing oscillator and, e) a strong nonlinear
oscillator. The results are given in plots, state variables vs. time, phase plots and Poincaré
maps. Neither divergence nor numerical damping was found for the chosen values of T.