Statistics of unstable periodic orbits of a chaotic dynamical system with a large number of degrees of freedom

For a simple model of chaotic dynamical systems with a large number of degrees of freedom, we find that there is an ensemble of unstable periodic orbits (UPOs) with the special property that the expectation values of macroscopic quantities can be calculated using only one UPO sampled from the ensemble. Evidence to support this conclusion is obtained by generating the ensemble by Monte Carlo calculation for a statistical mechanical model described by a space-time Hamiltonian that is expressed in terms of Floquet exponents of UPOs. This result allows us to interpret the recent interesting discovery that statistical properties of turbulence can be obtained from only one UPO [G. Kawahara and S. Kida, S. Kato and M. YamadaJ. Fluid Mech. 449, 291 (2001); , Phys. Rev. E 68, 025302(R) (2003)].

Figure 1

Averages (solid circles) and standard deviations (error bars) of ${\epsilon }_N$ evaluated from 50 000 samples of UPOs for the case $N=2^n(2⩽n⩽12)$, $k=1$, and $p=64$. The broken line represents the exact value in the limit $N\to \infty$, $-\tanh (1∕2)$.