Drawing the output of dynamical systems by juxtaposing local outputs

A dynamical system being described by its state equations and its initial state, we develop a method for drawing its output: It is based on the juxtaposition of local approximating outputs on successive time intervals $[t_i, t_(i+1)]0<=i<=n-1$. It consists in computing an approximated value of the state at initial point $t_i$ and an approximated output $y_i(t)$ on $[t_i, t_(i+1)]0<=i<=n-1$. An expression of the generating series $G_(q_r,t)$ for every component $q_r$ of the state $q$, an expression of the generating series $G_(y,t)$ of the output $y$ truncated at order $k$ are calculated and specified at every initial point $t_i$. We obtain an approximated output $y(t)$ at order $k$ in every interval $[t_i, t_(i+1)]0<=i<=n-1$. This method presents some theoretical advantages over Runge-Kutta methods: genericity, independency of the system and of the input, estimate of the error. So, an estimate of the suitable largest step can be computed. We have developed a Maple package for the creation of the generic expression of $G_(q_r,t),G_(y,t)$ and $y(t)$ at order $k$ and for the drawing of the local curves on every interval $[t_i, t_(i+1)]0<=i<=n-1$. For stable systems with oscillating output, for unstable systems near instability points, our method provides an appropriate result when a Runge-Kutta method is not suitable.