Trajectories--the issue of exponential divergence

In the hard-chaotic systems we consider, there is exponential divergence of initially-close trajectories [154]. We need to simulate the Hamiltonian dynamics using a computer with necessarily finite-precision arithmetic. This means that numerical round-off error will, after a few times the Lyapunov time $t_{\lambda}$, grow to dominate any trajectory, thereby destroying any relevance of the initial conditions. Might this be a problem for estimating correlation spectra of functions along such a trajectory? Especially one might worry about frequencies $\omega \ll t_{\lambda}^{-1}$, which are fundamental to the demonstration of the various $\omega\rightarrow0$ behaviours in Chapters 3 and 4.

It turns out that there is no cause for concern. One is saved by the very same exponential divergence, which forces all correlation functions to die exponentially like $e^{-t/t_{\lambda}}$. Therefore times separated by $\gg t_{\lambda}$ are effectively independent samples and do not contribute to small-frequency components in a spectrum. All the $\omega\rightarrow0$ behaviours in hard-chaos systems result from short time correlations less than or order $t_{\lambda}$, approximately the ergodic time. (This fact alone is sufficient to deduce the possible power laws--done in Appendix D).

This issue is slightly modified for chaotic systems with power-law correlation function decay, for instance the stadium (see Section 3.4). If correlations persist for long times, the Lyapunov time of contributing orbits must be equally long. Therefore these orbits will not become dominated by amplified round-off error noise, rather they will stay numerically accurate. In general, any trajectory can be computed accurately enough that round-off error does not invalidate its contribution to a correlation function.

It turns out that in hyperbolic chaotic systems, a numerically-generated `pseudo-trajectory' shadows (is very close to) a true trajectory with slightly different initial conditions [89,83]. However, due to the above arguments, this shadowing theorem is not necessary to give us confidence in correlation function estimates.