Comment on “Time-averaged properties of unstable periodic orbits and chaotic orbits in ordinary differential equation systems”

A recent paper claims that mean characteristics of chaotic orbits differ from the corresponding values averaged over the set of unstable periodic orbits, embedded in the chaotic attractor. We demonstrate that the alleged discrepancy is an artifact of the improper averaging. Since the natural measure is nonuniformly distributed over the attractor, different periodic orbits make different contributions into the time averages. As soon as the corresponding weights are accounted for, the discrepancy disappears.

Figure 1

(Color online) Time averages of $z\left(t\right)$ for chaotic trajectories and UPOs of the Lorenz equations. (a) Probability density for orbits of length $N=20$. (Solid line) Average values from ${10}^6$ segments of chaotic trajectories. (Dotted line) Weighted means over all UPOs with $N=20$. (Dashed line) Summation (without weights) over all UPOs with $N=20$. (b) Standard deviation of $⟨z⟩$ for periodic (pluses) and chaotic (crosses) orbits with $N$ turns. (Dashed lines) Power-law fits.