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Stochastic webs with Fourfold Rotation Symmetry

  • J. S. W. Lamb
Part of the NATO ASI Series book series (NSSB, volume 331)

Abstract

Weak chaos in nonlinear Hamilton systems has been studied extensively over the last two decades. A lot of attention was paid to the two-dimensional Chirikov-Taylor symplectic (i.e. area-preserving) standard map, modelling (part of) the motion of a charged particle in an inhomogeneous kicking electric field. This map displays weak chaos if it is perturbed from integrability (Chirikov, 1979; Lichtenberg and Lieberman, 1983). In the standard map, the chaotic motion is trapped between KAM-curves if the perturbation from integrability (here the strength of the electric field) is smaller than some threshold value. This causes a separation of chaotic regions in so-called stochastic layers. Only if the perturbation exceeds the threshold value, the chaotic motion is not trapped and the stochastic layers connect to form a two-dimensional infinite net of chaotic saddle connections, a so-called stochastic web, along which chaotic motion proceeds. The latter process is called stochastic diffusion. In the standard map, the threshold for stochastic diffusion is related to the break-up of the last KAM-curve.

Keywords

Unstable Manifold Saddle Connection Crystallographic Group Heteroclinic Connection Hyperbolic Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Afanasiev, V. Chernikov, A.A., Sagdeev, R.Z., and Zaslavsky, G.M., 1990, The width of the stochastic web and particle diffusion along the web, Phys. Lett. A 144:229.CrossRefMathSciNetADSGoogle Scholar
  2. Chernikov, A.A., Sagdeev, R.Z., and Zaslavsky, G.M., 1988, Stochastic webs, Physica D 33:65.CrossRefMATHMathSciNetADSGoogle Scholar
  3. Chirikov, B.V., 1979, A universal instability of many-dimensional oscillator systems, Phys. Rep. 216:63.MathSciNetGoogle Scholar
  4. Hoveijn, I., 1992, Symplectic reversible maps, tiles and chaos, Chaos, Solitons and Fractals 2:81.CrossRefMATHMathSciNetADSGoogle Scholar
  5. Lamb, J.S.W., 1992, Reversing symmetries in dynamical systems, J. Phys. A 25:925.CrossRefMATHMathSciNetADSGoogle Scholar
  6. Lamb, J.S.W., 1993, Crystallographic Symmetries of Stochastic Webs, J. Phys. A 26, 2921.CrossRefMATHMathSciNetADSGoogle Scholar
  7. Lamb, J.S.W., and Capel, H.W., 1993, Local bifurcations on the plane with reversing point group symmetry, Chaos, Solitons and Fractals to appear.Google Scholar
  8. Lamb, J.S.W., and Quispel, G.R.W., 1993, Reversing k-Symmetries in Dynamical Systems, preprint.Google Scholar
  9. Loeb, A.L., 1971, “Color and symmetry”, Wiley & Sons, New York.Google Scholar
  10. Ludwig, W., and Falter, C., 1988, “Symmetries in physics”, Springer Series in Solid-State Sciences, Vol. 64, Springer, Berlin.CrossRefMATHGoogle Scholar
  11. Lichtenberg, A.J., and Lieberman, M.A., 1983, “Regular and stochastic motion”, Applied Mathematical Sciences, Vol. 38, Springer, New York.CrossRefMATHGoogle Scholar
  12. Quispel, G.R.W., and Lamb, J.S.W., 1993, Dynamics and k-symmetries, this volume.Google Scholar
  13. 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.CrossRefMathSciNetADSGoogle Scholar
  14. Sevryuk, M.B., 1986, “Reversible systems”, Lecture Notes in Mathematics, Vol. 1211, Springer, Berlin.MATHGoogle Scholar
  15. 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.CrossRefGoogle Scholar
  16. Zaslavsky, G.M., Zakharov, M.Yu., Sagdeev, R.Z., Usikov, D.A., and Chernikov, A.A., 1986, Generation of ordered structures with a symmetry axis from a Hamiltonian dynamics, JETP Lett. 44:451 [Pis’ma Zh. Eksp. Teor. Fiz. 44:349], ibid., 1986, Stochastic web and diffusion of particles in a magnetic field, Sov. Phys. JETP 64:294 [Zh. Eksp. Teor. Fiz. 91:500].MathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • J. S. W. Lamb
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

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