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Dynamics and K-Symmetries

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Hamiltonian Mechanics

Part of the book series: NATO ASI Series ((NSSB,volume 331))

Abstract

The presence of symmetries is common in many dynamical systems of interest, endowing the system with certain simplifying features. A lot of attention has been given to continuous symmetries, since they give rise to conservation laws allowing one to reduce the number of degrees of freedom. During the past two decades attention has been directed to discrete symmetries as well, often in the context of bifurcation problems (Sattinger, 1979; Vanderbauwhede, 1982; Golubitsky et al., 1988).

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References

  • Chirikov, B.V., 1979, A universal instability of many-dimensional oscillator systems, Phys. Rep. 216:63.

    MathSciNet  Google Scholar 

  • Chossat, P., and Golubitsky, M., 1988, Symmetry-increasing bifurcation of chaotic attractors, Physica D 32:423.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • D’Ariano, G.M., Evangelista, L.R., and Saraceno, M., 1992, Classical and quantum structures in the kicked-top model, Phys.Rev.A 45:3646.

    Article  ADS  Google Scholar 

  • DeVogelaere, R., 1958, On the structure of symmetric periodic solutions of conservative systems, with applications, in: “Contributions to the Theory of Nonlinear Oscillations” Vol. 4, ed. S. Lefschetz, Princeton University Press, Princeton.

    Google Scholar 

  • Golubitsky, M., Stewart, I., and Schaeffer, D.G., 1988, “Singularities and Groups in Bifurcation Theory”, Vol. 2, Applied Mathematical Sciences, Vol. 69, Springer, New York.

    Book  MATH  Google Scholar 

  • Hoveijn, I., 1992, Symplectic reversible maps, tiles and chaos, Chaos, Solitons and Fractals 2:81.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Kimball, J.C., and Dumas, H.S., 1990, Symmetry in regular motion: relation to chaos, Phys. Lett. A 144:201.

    Article  MathSciNet  ADS  Google Scholar 

  • Lamb, J.S.W., 1992, Reversing symmetries in dynamical systems, J. Phys. A 25:925.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Lamb, J.S.W., 1993a, Crystallographic symmetries of stochastic webs, J. Phys. A 26 (in press).

    Google Scholar 

  • Lamb, J.S.W., 1993b, Stochastic webs with fourfold rotation symmetry, this volume.

    Google Scholar 

  • Lamb, J.S.W., and Quispel, G.R.W., 1993, Reversing k-Symmetries in Dynamical Systems, preprint.

    Google Scholar 

  • Llibre, J. and MacKay, R.S., 1992, Pseudo Anosov homeomorphisms on a sphere with four punctures have all periods, preprint.

    Google Scholar 

  • MacKay, R.S., 1984, Equivariant universality classes, Phys. Lett. A 106:99.

    Article  MathSciNet  ADS  Google Scholar 

  • Piña, E. and Jiménez Lara, L., 1987, On the symmetry lines of the standard mapping, Physica D 26:369.

    Article  MATH  ADS  Google Scholar 

  • Piña, E. and Cantoral, E., 1989, Symmetries of the quasi-crystal mapping, Phys. Lett. A 135:190.

    Article  MathSciNet  ADS  Google Scholar 

  • Ichikawa, Y.H., Kanimura, T., Hatori, T., and Kim, S.Y., 1989, Stochasticity and symmetry of the standard map, Prog. Theor. Phys. Supp. 98:1.

    Article  ADS  Google Scholar 

  • Richter, P.H., Scholz, H., and Wittek, A., 1990, A breathing chaos, Nonlinearity 3:45.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Roberts, J.A.G., and Baake, M., 1993, Trace maps as 3-d reversible dynamical systems with an invariant, preprint.

    Google Scholar 

  • Roberts, J.A.G., and Quispel, G.R.W., 1992, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep. 216:63.

    Article  MathSciNet  ADS  Google Scholar 

  • Sattinger, D.H., 1979, “Group Theoretic Methods in Bifurcation Theory”, Lecture Notes in Mathematics, Vol. 762, Springer, New York.

    MATH  Google Scholar 

  • Vanderbauwhede, A., 1982, “Local Bifurcation and Symmetry”, Research Notes in Mathematics, Vol. 75, Pitman, London.

    MATH  Google Scholar 

  • Zaslavsky. G.M., Sagdeev, R.Z., Usikov, D.A., and Chernikov, A.A., 1991 “Weak Chaos and Quasireg-ular Patterns”, Cambridge Nonlinear Science Series Vol.1, Cambridge University Press, Cambridge.

    Book  Google Scholar 

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© 1994 Springer Science+Business Media New York

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Quispel, G.R.W., Lamb, J.S.W. (1994). Dynamics and K-Symmetries. In: Seimenis, J. (eds) Hamiltonian Mechanics. NATO ASI Series, vol 331. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0964-0_30

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  • DOI: https://doi.org/10.1007/978-1-4899-0964-0_30

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4899-0966-4

  • Online ISBN: 978-1-4899-0964-0

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