### 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.

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.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**