Phase space error control for dynamical systems

Variable time-stepping algorithms for initial value ordinary differential e quations are traditionally designed to solve a problem for a fixed initial condition and over a finite time. It can be shown that these algorithms may perform poorly for long time computations with initial conditions that lie in a small neighborhood of a fixed point. In this regime there are orbits that are bounded in space but unbounded in time, and the classical error-pe r-step or error-per-unit-step philosophy may be improved upon. A new error criterion is introduced that essentially bounds the truncation error at eac h step by a fraction of the solution arc length over the corresponding time interval. This new control can be incorporated within a standard algorithm as an additional constraint at negligible additional computational cost. I t is shown that this new criterion has a positive effect on the linear stab ility properties and hence improves behavior in the neighborhood of stable fixed points. Furthermore, spurious fixed points and period two solutions a re prevented. The new criterion is shown to be admissible in the sense that it can always be satisfied with nonzero stepsizes. Implementation details and numerical results are given.