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A Straightforward Approach To Solve Ordinary Nonlinear Differential Systems

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 Poincare 
maps. Neither divergence nor numerical damping was found for the chosen values of T.