The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems. In particular, we examine, quantitatively, the closeness of the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in the set. We also analyze the difference between the long-time average values of physical quantities evaluated with respect to a dense trajectory and those computed from unstable periodic orbits. Results with both the Hénon map and the Ikeda–Hammel–Jones–Moloney map for which periodic orbits can be enumerated lend credence to the conjecture that the unstable periodic-orbit theory of the natural measure is applicable to nonhyperbolic chaotic saddles.
6 Figures and Tables
Fig. 1. A schematic illustration of a cell in the Markov partition defined with respect to the stable and the unstable manifolds of a hyperbolic chaotic saddle.
Fig. 3. (a–d) For the Hénon map at a = 1.5, 1.55, 1.6 and 1.65, respectively, lnµS(p) versus p, where µS(p) is the total measure represented by all periodic orbits of period p defined in Eq. (10).
Fig. 4. Comparison of the lifetimes of chaotic saddle obtained by extracting the slopes of the lines of lnN(n) versus n (circles) and these obtained via Eq. (10) through unstable periodic orbits (diamonds) for a = 1.5, 1.55, 1.6 and 1.65.
Fig. 5. (a–d) For the Hénon map at a = 1.5, 1.55, 1.6 and 1.65, respectively, ln ∆µ(p) versus p. We see that ∆µ(p) decreases exponentially as p increases, indicating the applicability of Eq. (9) for large periods.
Fig. 6. (a–d) For the Hénon map at a = 1.5, 1.55, 1.6 and 1.65, respectively, 〈F 〉 (dotted line) and 〈F (p)〉 (large dots) versus p, where F (x, y) = ex 2+y2 . It shows that 〈F (p)〉 converges to 〈F 〉 as p is increased.
Fig. 10. For the IHJM map, 〈F 〉 (dotted line) and 〈F (p)〉 (continuous line with dots) versus p, where F (x, y) = ex2+y2 . Again, 〈F (p)〉 converges to 〈F 〉 as p is increased.
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