Several long runs have been done (and are being done) up to now. They require a massive simulation effort. We would like to report on the experimental behavior of HIDRA on several problems.
A systematic and massive computation of Lyapunov exponents in that simple problem allows to understand some key points in the reducibility problem for quasiperiodic coefficients and in the breakdown of tori. HIDRA allows to compute the time 2p-map at a rate of more than 5,000,000 iterates per second. See figure 1 for the resonance tongues and the line of tori breakdown.
Many dynamical systems have to be simulated in an interactive way to gain insight on the dynamical properties. HIDRA allows to obtain between 104 and 105 Poincaré iterates (through a suitable transversal section) per second on simple ordinary differential equations of moderate dimension (say, between 3 and 7). This high performance allows quick interactive explorations. An invariant manifold like the one whose section is displayed in figure 2, related to the unfolding of a codimension 3 bifurcation, is computed by HIDRA in 5 seconds.
In all the computations mentioned above the sustained output has been, at least, 2 Gflops, reaching 2.7 Gflops in processes with low rates of transfer of data. (September, 1998).
Figure 1: Resonance tongues and line of collapse of resonances, where the breakdown of tori occurs.
Figure 2: Section of a 2D invariant unstable manifold of a 3D system.