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

We point out that the original work [Saiki and Yamada, Phys. Rev. E 79, 015201(R) (2009)] was aimed at addressing the question of why even a single periodic orbit of low period can capture averaged properties of physical quantities reasonably well in some dynamical systems including turbulent fluids. This point of view is quite different from that of the Comment [Zaks and Goldobin, Phys. Rev. E 81, 018201 (2010)].

Figure 1

(Color online) Var of $z$ vs. $⟨z⟩$ for the Lorenz system [(a) Chaotic segments $\left(N=11\right)$ [dot (green online)], (b) UPO $\left(N=11\right)$ [small cross (red online)], and Chaos [big cross (blue online)].