Periodic orbits

    Periodic orbits are states that exactly repeat themselves after a certain time. They are often dense, so any state of the system is very close to many periodic orbits, even if the probability of exact periodicity is zero. This means they can still be used to approximate the dynamics. For chaotic systems, the periodic orbit theory gives systematic expansions of desires statistical properties such as averages, correlation functions, Lyapunov exponents and dimensions in terms of the unstable periodic orbits. I have worked to generalise the theory to intermittent dynamics, stochastically perturbed dynamics and spatiotemporal dynamics. Periodic orbits play a different, but equally vital role in integrable systems.

  1. Stability ordering of cycle expansions C. P. Dettmann and G. P. Morriss, Phys. Rev. Lett. 78, 4201-4204 (1997) pdf ps arxiv
  2. Irreversibility, diffusion and multifractal measures in thermostatted systems, C. P. Dettmann, G. P. Morriss, and L. Rondoni, Chaos, Solitons and Fractals 8, 783-792 (1997)
  3. Recent results for the thermostatted Lorentz gas, G. P. Morriss, C. P. Dettmann and L. Rondoni, Physica A 240, 84-95 (1997)
  4. Cycle expansions for intermittent diffusion C. P. Dettmann and P. Cvitanovic', Phys. Rev. E 56, 6687-6692 (1997) pdf ps arxiv
  5. Computing the diffusion coefficient for intermittent maps: Resummation of stability ordered cycle expansions C. P. Dettmann and P. Dahlqvist, Phys. Rev. E 57, 5303-5310 (1998) pdf ps.gz arxiv
  6. Trace formulas for stochastic evolution operators: Weak noise perturbation theory P. Cvitanovic', C. P. Dettmann, R. Mainieri, and G. Vattay, J. Stat. Phys. 93, 981-999 (1998) ps.gz (1.2M when uncompressed) arxiv
  7. Traces and determinants of strongly stochastic operators C. P. Dettmann, Phys. Rev. E 59, 5231-5234 (1999) pdf ps html arxiv
  8. Trace formulas for stochastic evolution operators: Smooth conjugation method P. Cvitanovic', C. P. Dettmann, R. Mainieri, and G. Vattay, Nonlinearity 12, 939-953 (1999) ps arxiv
  9. Spectrum of stochastic evolution operators: Local matrix representation approach P. Cvitanovic', N. Sondergaard, G. Palla, G. Vattay, and C. P. Dettmann, Phys. Rev. E 60, 3936-3941 (1999) pdf ps Two distinct arxiv versions:arxiv arxiv
  10. Microscopic chaos and diffusion C. P. Dettmann and E. G. D. Cohen, J. Stat. Phys. 101, 775-817 (2000) ps.gz (28 pages; 2.1M when uncompressed) arxiv
  11. Noise corrections to stochastic trace formulas G. Palla, G. Vattay, A. Voros, N. Sondergaard, C. P. Dettmann, Found. Phys. 31, 641-657 (2001). arxiv
  12. Stable synchronised states of coupled Tchebyscheff maps, C. P. Dettmann, Physica D 172 88-102 (2002). ps.gz(1.3M when uncompressed) animation (0.3M) arxiv
  13. Lyapunov spectra of periodic orbits for a many-particle system, T. Taniguchi, C. P. Dettmann and G. P. Morriss, J. Stat. Phys. 109 747-764 (2002). pdf ps.gz(1.0M when uncompressed). arxiv
  14. Fractal asymptotics, C. P. Dettmann, Physica D 187, 214-222 (2004). ps arxiv
  15. Periodic orbit theory of two coupled Tchebyscheff maps, C. P. Dettmann and D. Lippolis, Chaos, Solitons and Fractals 23 43-54 (2005). ps pdf arxiv
  16. Asymptotic expansions for the escape rate of stochastically perturbed unimodal maps, C. P. Dettmann and T. B. Howard, Physica D, 238, 2404-2408 (2009). pdf arxiv early poster
  17. Open mushrooms: Stickiness revisited, C. P. Dettmann and O. Georgiou, J. Phys. A.: Math. Theor. 44 195102 (2011). [Highlighted in a JPA Insights article.] pdf arxiv poster
  18. Dependence of chaotic diffusion on the size and position of holes, G. Knight, O. Georgiou, C. P. Dettmann, R. Klages, Chaos 22 023132 (2012). pdf arxiv
  19. Faster than expected escape for a class of fully chaotic maps, O. Georgiou, C. P. Dettmann, E. G. Altmann, Chaos 22 043115 (2012). arxiv pdf
  20. Open circle maps: Small hole asymptotics, C. P. Dettmann, Nonlinearity 26 307-317 (2013). pdf arxiv
  21. Book chapter: The Lorentz gas as a paradigm for nonequilibrium stationary states, C. P. Dettmann, pp 315-365 in Hard ball systems and the Lorentz gas (edited by D. Szasz), Encyclopaedia of Mathematical Sciences Vol 101 (Springer, 2000). Full size version, 50 pages ps. Environmental microscopic version, 25 pages ps.
  22. Web book contribution: "Stability ordering of cycle expansions" (currently section 20.5) in Classical and Quantum Chaos webbook, P. Cvitanovic' et al html
  23. Poster: Recent developments in periodic orbit theory C. P. Dettmann April 2002. ps
  24. Poster: Averaging quasiperiodic systems, C. P. Dettmann and T. B. Howard, August 2008 pdf

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