Accurate computation of chaotic dynamical systems
Walter Krämer · 2007
The computation of orbits of dynamical systems is known to be highly unstable if the system exhibits chaotic behavior. In this case, even for the very simplest systems, ordinary floating-point computations will eventually deliver results which are completely wrong quantitatively, when compared with the true trajectory on which the computation began. Similarly, ordinary interval arithmetic (i. e. intervals of floating-point numbers) yield poor enclosures after few iterations. In most cases the computation breaks down because of overflow. Using intpakX's multiple precision intervals, we can compute enclosures of orbits for a considerably longer time with high accuracy. Statements concerning the sensitivity with respect to small changes in the seed value of the numerical computations are possible with mathematical rigor. We also show that computing an orbit using a rational arithmetic as e.g. provided by the computer algebra system Maple is not possible due to computing time and computer memory limitations.