  Recent advances in nonlinear dynamical systems theory provide a new insight
into numerical properties of discrete algorithms developed to solve nonlinear
initial value problems. Basic features like accuracy and stability are well
pointed out through diagrams or maps of computed asymptotic solutions in a
suitable parametric space. Applying this methodology to a nonlinear test
equation, we compared some numerical features of the well known second-order
Crank-Nicolson solver with those of a recent proposed version which is
fourth-order accurate. The approach gives some useful indication on the
capabilities of familiar and innovative ODE integrators when applied to
nonlinear problems.
