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

Abstract

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, J. Fluid Mech. {\bf 449}, 291 (2001); S. Kato and M. Yamada, Phys. Rev. E {\bf 68}, 025302(R)(2003)].

4 pages, 1 figure. In order to clarify generality of our result and the role of a large number of degrees of freedom, a brief subsection was added

References in corpus (3)

• Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns
• Variational method for finding periodic orbits in a general flow
• Analytical Approach for Piecewise Linear Coupled Map Lattices

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