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Symmetry Breaking in the Period Doubling Route to Chaos

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Book cover Chaotic Dynamics

Part of the book series: NATO ASI Series ((NSSB,volume 298))

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Abstract

The period doubling route to chaos is not quite as universal as one is often made to believe. In many practical situations the well known sequence 1 → 2 → 4 → 8 …. → chaos is interrupted or even completely broken off before chaos is reached [1–6]. The present paper is about one particular (rather mild) kind of interruption: a symmetry breaking bifurcation, already at the level of period 2. The period 2 solution, instead of period doubling to a period 4 solution, bifurcates into two non-symmetric solutions of period 2. Subsequently both of the two newly formed period 2 solutions resume the period doubling route as if nothing had happened. The complete scenario can thus be described by the following sequence: 1→2→2x (2→4→8→16→… → chaos).

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References

  1. V. Franceschini, “Bifurcations of tori and phase locking in a dissipative system of differential equations”, Physica 6D:285–304 (1983).

    MathSciNet  Google Scholar 

  2. E. Knobloch and N.O. Weiss, “Bifurcations in a model of magnetoconvection”, Physica 9D:379–407 (1983)

    MathSciNet  Google Scholar 

  3. G.L. Oppo and A. Politi, “Collision of Feigenbaum cascades”, Phys. Rev. A30:435–441 (1984).

    MathSciNet  Google Scholar 

  4. M. Bier and T.C. Bountis, “Remerging Feigenbaum trees in dynamical systems”, Phys. Lett. 104A:239–244 (1984).

    MathSciNet  Google Scholar 

  5. T.P. Valkering and J.H.J. van Opheusden, Bouncing ball behaviour of a solitary wave, in: “Proc. of the conference on Nonlinear Dynamics”, Bologna, Italy, G. Turchetti, ed., World Scientific, Singapore, 225–235 (1988).

    Google Scholar 

  6. Th. Zeegers, “On the existence of infinite period doubling sequences in a class of 4D semi-symplectic mappings”, J. Phys. A: Math. Gen. 24:2287–2314 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  7. R.M. May, “Bifurcations and dynamic complexity in ecological systems”, Annals of the New York Academy of Sciences 316:517–529 (1979).

    Article  Google Scholar 

  8. R.M. May, “Nonlinear phenomena in ecology and epidemiology”, in: Nonlinear Dynamics, edited by R.H.G. Helleman, Annals of the New York Academy of Sciences 357:267–281 (1980).

    Google Scholar 

  9. L.D. Pustyl’nikov, “Stable and oscillating motions in nonautonomous dynamical systems II” Trans. Moscow Math Soc. 2:1–101 (1978).

    Google Scholar 

  10. P.J. Holmes, “The dynamics of repeated impacts with a sinusoidally vibrating table”, J. Sound Vib. 84:173–189 (1982).

    MATH  Google Scholar 

  11. R.M. Everson, “Chaotic dynamics of a bouncing ball”, Physica 19D:355–383 (1986).

    MathSciNet  Google Scholar 

  12. P. Pieranski, “Jumping particle model. Period doubling cascade in an experimental system”, J. Physique 44:573–578 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Pieranski, Z.J. Kowalik and M. Franaszek, “Jumping particle model. A study of the phase space of a non-linear dynamical system below its transition to chaos”, J. Physique 46:681–686 (1985).

    Article  Google Scholar 

  14. P. Pieranski and R. Bartolino, “Jumping particle model. Modulation modes and resonant response to a periodic perturbation, J. Physique 46:687–690 (1985).

    Article  Google Scholar 

  15. M. Franaszek and Z.J. Kowalik, “Measurements of the dimension of the strange attractor for the Fermi-Ulam problem”, Phys. Rev. A33:3508–3510 (1986).

    Google Scholar 

  16. M. Franaszek and P. Pieranski, “Jumping particle model. Critical slowing down near the bifurcation points”, Can. J. Phys. 63:488–493 (1985).

    Article  Google Scholar 

  17. N.B. Tufillaro, T.M. Mello, Y.M. Choi and A.M. Alabano, “Period doubling boundaries of a bouncing ball”, J. Physique 47:1477–1482 (1986).

    Article  Google Scholar 

  18. N.B. Tufillaro and A.M. Alabano, “Chaotic dynamics of a bouncing ball”, Am. J. Phys. 54:939–944 (1986).

    Article  Google Scholar 

  19. T.M. Mello and N.B. Tufillaro, “Strange attractors of a bouncing ball”, Am. J. Phvs. 55:316–320 (1987).

    Google Scholar 

  20. K. Wiesenfeld and N.B. Tufillaro, “Suppression of period doubling in the dynamics of a bouncing ball”, Physica 26D:321–335 (1987).

    Google Scholar 

  21. R. Abraham and J.E. Marsden, “Foundations of mechanics”, 2nd edition Benjamin Cummings, Reading Mass., (1978); the “symmetric saddle-node bifurcation” is discussed on page 504.

    Google Scholar 

  22. R.S. MacKay, “The dominant symmetry of reversible maps”, in his Ph.D. thesis, Princeton University (1982); here the bifurcation is named after R.J. Rimmer, J. Diff. Eqns. 29:329 (1978).

    Google Scholar 

  23. E.A. Jackson, “Perspectives of nonlinear dynamics”, Cambridge University Press, Cambridge, vol. 1, (1989); the “double point bifurcation” is discussed in chapter 3.2.

    Google Scholar 

  24. 24. T.C. Bountis, “Period doubling bifurcations and universality in conservative systems”, Physica 3D:577–589 (1981).

    MathSciNet  Google Scholar 

  25. J.M. Greene, R.S. MacKay, F. Vivaldi and M.J. Feigenbaum, “Universal behaviour in families of area-preserving maps”, Physica 3D:468–486 (1981).

    MathSciNet  Google Scholar 

  26. M.J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations”, J. Stat. Phvs. 19:25–52 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  27. J.P. van der Weele, H.W. Capel, T. Post and Ch.J. Calkoen, “Crossover from dissipative to conservative behaviour in period doubling systems”, Physica 137A:1–43 (1986).

    Google Scholar 

  28. L.D. Landau and E.M. Lifschitz, “Mechanics”, volume 1 of Course of Theoretical Physics, Pergamon Press, Oxford, 1960; the parametrically driven pendulum is treated in the exercises 5.3c, 27.3 and 30.1.

    Google Scholar 

  29. J.B. MacLaughlin, “Period-doubling bifurcations and chaotic motion for a parametrically forced pendulum”, J. Stat. Phys. 24:375–388 (1981).

    Article  Google Scholar 

  30. R.W. Leven and B.P. Koch, “Chaotic behaviour of a parametrically excited pendulum”, Phvs. Lett. 86A:71–74 (1981).

    Article  Google Scholar 

  31. B.P. Koch and R.W. Leven, “Subharmonic and homoclinic bifurcations in a parametrically forced pendulum”, Physica 16D:1–13 (1985).

    MathSciNet  Google Scholar 

  32. J.P. van der Weele and T.P. Valkering, Orde en chaos in de parametrisch aangedreven slinger (in Dutch), in: “Dynamische Systemen en Chaos”, H.W. Broer and F. Verhulst, eds., Epsilon, Utrecht (1990), 232–255.

    Google Scholar 

  33. W. Case, “Parametric instability: an elementary demonstration and discussion”, Am. J. Phys. 48:218–221 (1980).

    Article  Google Scholar 

  34. B.P. Koch, R.W. Leven, B. Pompe and C. Wilke, “Experimental evidence for chaotic behaviour of a parametrically forced pendulum” Phys. Lett. 96A:219-224 (1983); an experimental picture of the “symmetry breaking bifurcation” is presented on page 222.

    Google Scholar 

  35. R.W. Leven, B. Pompe, C. Wilke and B.P. Koch, “Experiments on periodic and chaotic motions of a parametrically forced pendulum”, Phvsica 16D:371–384 (1985).

    MathSciNet  Google Scholar 

  36. W. van de Water and M. Hoppenbrouwers, Chaos and the experiment, in: ”Proceedings of the Spring Meeting on Nonlinear Dynamics Twente”, University of Twente, Enschede, March (1990) 58–79.

    Google Scholar 

  37. R.H.G. Helleman, with an appendix by R.S. MacKay, One mechanism for the onset of large-scale chaos in conservative and dissipative systems, in: “Long-time prediction in dynamics”, C.W. Horton Jr., L.E. Reichl and A.G. Szebehely, eds., Wiley and Sons, New York, (1985) 95–126.

    Google Scholar 

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Authors and Affiliations

  1. Center for Theoretical Physics, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    J. P. van der Weele

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  1. J. P. van der Weele
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Editors and Affiliations

  1. University of Patras, Patras, Greece

    T. Bountis

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© 1992 Springer Science+Business Media New York

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van der Weele, J.P. (1992). Symmetry Breaking in the Period Doubling Route to Chaos. In: Bountis, T. (eds) Chaotic Dynamics. NATO ASI Series, vol 298. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3464-8_33

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  • DOI: https://doi.org/10.1007/978-1-4615-3464-8_33

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6534-1

  • Online ISBN: 978-1-4615-3464-8

  • eBook Packages: Springer Book Archive

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