Statistical Degradation of Chaotic Dynamical Systems on Computers
Speaker: Ross Ah-Weng
Abstract: Many chaotic physical systems are forecasted using Birkhoff averages of observables along numerically simulated trajectories. However, when any dynamical system is simulated in floating-point arithmetic, the continuum is replaced by a finite discrete domain. The new dynamics are substantially different: every trajectory is eventually periodic, sufficiently close trajectories become indistinguishable after roundoff, and any invariant measure is supported on a finite set of disjoint periodic orbits. We identify a generic mechanism in which non-injectivity from rounding causes statistics inferred from any simulated trajectory to continually and irreversibly degrade. We develop this framework using the formalism of functional graphs, the Perron-Frobenius Operator, and derive general error bounds on observables. We validate this framework numerically on the Logistic map and various parameters of the Generalised Bernoulli Map, at increasing precision up to IEEE 754 FP32. In the case of stochastic rounding, we construct the associated Markov Chain and comment on a failure mode associated to exactly representable periodic orbits. Finally, we empirically test convergence properties from an initial uniform distribution, and find evidence that Birkhoff averages over finite timespans on computers consistently outperform the infinite time limit.