Tunable Orbits Influence in a Driven Stadium-Like Billiard

Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 22)

Abstract

The dynamics of a driven stadium-like billiard is investigated through a four-dimensional nonlinear mapping. We set a critical resonance velocity, which plays the role of an ensemble separation according to the initial velocities. When the resonance is active, the invariant curves that surround the stability islands become stochastic layers, thus allowing a change in the dynamics from chaos to stability and vice versa, leading the root mean square velocity to steady state plateaus for long times. A tunneling behavior of orbits in the lower ensemble was characterized via transport analysis and Lyapunov exponents. Our results may be extended to other similar dynamical systems that may present similar critical resonances.

Notes

Acknowledgements

ALPL acknowledges FAPESP (2014/25316-3) and FAPESP (2015/26699-6) for financial support. ALPL also thanks the University of Bristol for the kind hospitality during his stay in UK. This research was supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP). The author also acknowledges Alexander Loskutov (in memorian) for the art of Fig. 4.1.

References

  1. 1.
    Hilborn, R. C. (1994). Chaos and nonlinear dynamics: An introduction for Scientists and Engineers. Oxford: Oxford University Press.Google Scholar
  2. 2.
    Lichtenberg, A. J., & Lieberman, M. A. (1992). Regular and chaotic dynamics. Applied mathematical science (Vol. 38). Berlin: Springer Verlag.CrossRefGoogle Scholar
  3. 3.
    Zaslasvsky, G. M. (2007). Physics of chaos in Hamiltonian systems. New York: Imperial College Press.Google Scholar
  4. 4.
    Zaslasvsky, G. M. (2008). Hamiltonian chaos and fractional dynamics. Oxford: Oxford University Press.Google Scholar
  5. 5.
    Altmann, E. G., Portela, J. S. E., & Tél, T. (2013). Leaking chaotic systems. Reviews of Modern Physics, 85, 869.CrossRefGoogle Scholar
  6. 6.
    Meiss, J. D. (2015). Thirty years of turnstiles and transport. Chaos, 25, 097602.CrossRefGoogle Scholar
  7. 7.
    Leine, R. I., & Nijmeijer, H. (2013). Dynamics and bifurcations of non-smooth mechanical systems (Vol. 18). Berlin: Springer Science & Business.Google Scholar
  8. 8.
    Solomon, T. H., Weeks, E. R., & Swinney, H. L. (1993). Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. Physical Review Letters, 71, 3975.CrossRefGoogle Scholar
  9. 9.
    del-Castillo-Negrete, D., Carreras, B. A., & Lynch, V. E. (2005). Nondiffusive transport in plasma turbulence: A fractional diffusion approach. Physical Review Letters, 94, 065003.Google Scholar
  10. 10.
    Portela, J. S. E., Caldas, I. L., & Viana, R. L. (2007). Fractal and wada exit basin boundaries in tokamaks. International Journal of Bifurcation and Chaos, 17, 4067.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jimenez, G. A., & Jana, S. C. (2007). Electrically conductive polymer nanocomposites of polymethylmethacrylate and carbon nanofibers prepared by chaotic mixing. Composites: Part A 38, 983.CrossRefGoogle Scholar
  12. 12.
    He, P., Ma, S., & Fan, T. (2013). Finite-time mixed outer synchronization of complex networks with coupling time-varying delay. Chaos, 22, 043151.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Andersen, M. F., Kaplan, A., Grn̆zweig, T., & Davidson, N. (2006). Decay of quantum correlations in atom optics billiards with chaotic and mixed dynamics. Physical Review Letters, 97, 104102.Google Scholar
  14. 14.
    Abraham, N. B., & Firth, W. J. (1990). Overview of transverse effects in nonlinear-optical systems. Journal of the Optical Society of America B, 7(6), 951–962 (1990). https://doi.org/10.1364/JOSAB.7.000951.CrossRefGoogle Scholar
  15. 15.
    Milner, V., Hanssen, J. L., Campbell, W. C., & Raizen, M. G. (2001). Optical billiards for atoms. Physical Review Letters, 86, 1514.CrossRefGoogle Scholar
  16. 16.
    Altmann, E. G. (2009). Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics. Physical Review A, 79, 013830.CrossRefGoogle Scholar
  17. 17.
    Chernov, N., & Markarian, R. (2006). Chaotic billiards (Vol. 127). Providence: American Mathematical Society.MATHGoogle Scholar
  18. 18.
    Birkhoff, G. D. (1927). Dynamical systems. Providence: American Mathematical Society.CrossRefGoogle Scholar
  19. 19.
    Sinai, Y. G. (1970). Dynamical systems with elastic reflections. Russian Mathematical Surveys, 25, 137.CrossRefGoogle Scholar
  20. 20.
    Bunimovich, L. A. (1979). On the ergodic properties of nowhere dispersing billiards. Communications in Mathematical Physics, 65, 295.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Bunimovich, L. A., & Sinai, Y. G. (1981). Statistical properties of lorentz gas with periodic configuration of scatterers. Communications in Mathematical Physics, 78, 479.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gallavotti, G., & Ornstein, D. S. (1974). Billiards and bernoulli schemes. Communications in Mathematical Physics, 38, 83.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tanner, G., & Søndergaard, N. (2007). Wave chaos in acoustics and elasticity. Journal of Physics A, 40, 443.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Stein, J., & Støckmann, H. J. (1992). Experimental determination of billiard wave functions. Physical Review Letters, 68, 2867.CrossRefGoogle Scholar
  25. 25.
    Sirko, L., Koch, P. M., & Blümel, R. (1997). Experimental identification of non-newtonian orbits produced by ray splitting in a dielectric-loaded microwave cavity. Physical Review Letters, 78, 2940.CrossRefGoogle Scholar
  26. 26.
    Haake, F. (2001). Quantum signatures of chaos. Berlin: Springer.CrossRefGoogle Scholar
  27. 27.
    Ponomarenko, L. A., Schedin, F., Katsnelson, M. I., Yang, R., Hill, E. W., Novoselov, K. S., et al. (2008). Chaotic dirac billiard in graphene quantum dots. Science, 320, 356.CrossRefGoogle Scholar
  28. 28.
    Berggren, K. F., Yakimenko, I. I., & Hakanen, J. (2010). Modeling of open quantum dots and wave billiards using imaginary potentials for the source and the sink. New Journal of Physics, 12, 073005.CrossRefGoogle Scholar
  29. 29.
    Jalabert, R. A., Stone, A. D., & Alhassidd, Y. (1992). Statistical theory of Coulomb blockade oscillations: Quantum chaos in quantum dots. Physical Review Letters, 68, 3468.CrossRefGoogle Scholar
  30. 30.
    Meza-Montes, L., & Ulloa, S. E. (1997). Dynamics of two interacting particles in classical billiards. Physical Review E, 55, R6319.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Xavier, E. P. S., Santos, M. C., Dias da Silva, L. G. G. V., da Luz, M. G. E., & Beims, M. W. (2004). Quantum chaos for two interacting particles confined to a circular billiard. Physica A, 342, 377.CrossRefGoogle Scholar
  32. 32.
    Oliveira, H. A., Manchein, C., & Beims, M. W. (2008). Soft wall effects on interacting particles in billiards. Physical Review E, 78, 046208.CrossRefGoogle Scholar
  33. 33.
    Zharnitsky, V. (1995). Quasiperiodic motion in the billiard problem with a softened boundary. Physical Review Letters, 75, 4393.CrossRefGoogle Scholar
  34. 34.
    Fré, P., & Sorin, A. S. (2010). Supergravity black holes and billiards and the Liouville integrable structure associated with Borel algebras. Journal of High Energy Physics, 3, 1.MathSciNetMATHGoogle Scholar
  35. 35.
    Stone, A. D. (2010). Nonlinear dynamics: Chaotic billiard lasers. Nature, 465, 696.CrossRefGoogle Scholar
  36. 36.
    Bunimovich, L. A. (1974). On ergodic properties of certain billiards. Functional Analysis and Its Applications, 8, 73.MathSciNetCrossRefGoogle Scholar
  37. 37.
    Livorati, A. L. P., Loskutov, A., & Leonel, E. D. (2011). A family of stadium- like billiards with parabolic boundaries under scaling analysis. Journal of Physics A, 44, 175102.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Loskutov, A., & Ryabov, A. (2002). Particle dynamics in time-dependent stadium-like billiards. Journal of Statistical Physics, 108, 995.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Loskutov, A., Ryabov, A. B., & Leonel, E. D. (2010). Separation of particles in time-dependent focusing billiards. Physica A, 389, 5408.MathSciNetCrossRefGoogle Scholar
  40. 40.
    Markarian, R., Kamphorst, S. O., & de Carvalho, S. P. (1996). Chaotic properties of the elliptical stadium. Communications in Mathematical Physics, 174, 661.MathSciNetCrossRefGoogle Scholar
  41. 41.
    Loskutov, A., Ryabov, A. B., & Akinshin, L. G. (1999). Mechanism of Fermi acceleration in dispersing billiards with time-dependent boundaries. Journal of Experimental and Theoretical Physics, 89, 966.CrossRefGoogle Scholar
  42. 42.
    Loskutov, A., Ryabov, A. B., & Akinshin, L. G. (2000). Properties of some chaotic billiards with time-dependent boundaries. Journal of Physics A, 33, 7973.MathSciNetCrossRefGoogle Scholar
  43. 43.
    Fermi, E. (1949). On the origin of the cosmic radiation. Physical Review, 75, 1169.CrossRefGoogle Scholar
  44. 44.
    Lenz, F., Diakonos, F. K., & Schmelcher, P. (2008). Tunable fermi acceleration in the driven elliptical billiard. Physical Review Letters, 100, 014103.CrossRefGoogle Scholar
  45. 45.
    Lenz, F., Petri, C., Koch, F. R. N., Diakonos, F. K., & Schmelcher, P. (2009). Evolutionary phase space in driven elliptical billiards. New Journal of Physics, 11, 083035.CrossRefGoogle Scholar
  46. 46.
    Leonel, E. D., & Bunimovich, L. A. (2010). Suppressing fermi acceleration in a driven elliptical billiard. Physical Review Letters, 104, 224101.CrossRefGoogle Scholar
  47. 47.
    Livorati, A. L. P., Caldas, I. L., & Leonel, E. D. (2012). Decay of energy and suppression of Fermi acceleration in a dissipative driven stadium-like billiard. Chaos, 22, 026122.MathSciNetCrossRefGoogle Scholar
  48. 48.
    Livorati, A. L. P., Loskutov, A., & Leonel, E. D. (2012). A peculiar Maxwell’s Demon observed in a time-dependent stadium-like billiard. Physica A, 391, 4756.MathSciNetCrossRefGoogle Scholar
  49. 49.
    Livorati, A. L. P., Palmero, M. S., Dettmann, C. P., Caldas, I. L., & Leonel, E. D. (2014). Separation of particles leading either to decay or unlimited growth of energy in a driven stadium-like billiard. Journal of Physics A, 47, 365101.MathSciNetCrossRefGoogle Scholar
  50. 50.
    Karlis, A. K., Papachristou, P. K., Diakonos, F. K., Constantoudis, V., & Schmelcher, P. (2006). Hyperacceleration in a stochastic Fermi-Ulam model. Physical Review Letters, 97, 194102.CrossRefGoogle Scholar
  51. 51.
    Livorati, A. L. P., Ladeira, D. G., & Leonel, E. D. (2008). Scaling investigation of Fermi acceleration on a dissipative bouncer model. Physical Review E, 78, 056205.CrossRefGoogle Scholar
  52. 52.
    Diaz-I, G., Livorati, A. L. P., & Leonel, E. D. (2016). Statistical investigation and thermal properties for a 1-D impact system with dissipation. Physics Letters A, 380, 1830.CrossRefGoogle Scholar
  53. 53.
    Livorati, A. L. P., de Oliveira, J. A., Ladeira, D. G., & Leonel, E. D. (2014). Time-dependent properties in two-dimensional and Hamiltonian mappings. The European Physical Journal Special Topics, 223, 2953.CrossRefGoogle Scholar
  54. 54.
    Livorati, A. L. P., Dettmann, C. P., Caldas, I. L., & Leonel, E. D. (2015). On the statistical and transport properties of a non-dissipative Fermi-Ulam model. Chaos, 25, 103107.MathSciNetCrossRefGoogle Scholar
  55. 55.
    Wolf, A., Swift, J. B., Swinney, H. L., & Vastano, J. A. (1985). Determining Lyapunov exponents from a time series. Psysica D, 285, 16.MATHGoogle Scholar
  56. 56.
    Szezech, J. D. J., Lopes, S. R., & Viana, R. L. (2005). Finite-time Lyapunov spectrum for chaotic orbits of non-integrable Hamiltonian systems. Physics Letters A, 335, 394.MathSciNetCrossRefGoogle Scholar
  57. 57.
    Manchein, C., Beims, M. W., & Rost, J. M. (2014). Characterizing weak chaos in nonintegrable Hamiltonian systems: The fundamental role of stickiness and initial conditions. Physica A, 400, 186.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de FísicaUNESP - Univ Estadual PaulistaRio ClaroBrazil
  2. 2.School of MathematicsUniversity of BristolBristolUK

Personalised recommendations