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Structurally stable heteroclinic cycles in a system with O(3) symmetry

Part of the Lecture Notes in Mathematics book series (LNM,volume 1463)

Abstract

The existence and stability of structurally stable heteroclinic cycles are discussed in a codimension 2 bifurcation problem with O(3)-symmetry, when the critical spherical modes 1=1 and 1=2 occur at the same time. Several types of heteroclinic cycles are found, which may explain aperiodic attractors found in numerical simulations of the onset of convection in a self-gravitating fluid spherical shell (Friedrich, Haken [1986]).

Keywords

  • Phase Portrait
  • Bifurcation Diagram
  • Unstable Manifold
  • Strange Attractor
  • Pure Mode

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References

  • Armbruster D. Chossat P [1989]. Structurally stable heteroclinic cycles in mode interaction with O(3)-symmetry, to appear.

    Google Scholar 

  • Armbruster D, Guckenheimer J, Holmes P [1988]. Heteroclinic cycles and modulated traveling waves in systems with O(2) symmetry, Physica 29D, 257–282.

    Google Scholar 

  • Chossat P [1982]. Le problème de Bénard dans une couche sphérique, thesis, Université de Nice.

    Google Scholar 

  • Chossat P [1983]. Intéraction entres bifurcations de modes 1=1 et 1=2 dans les problèmes invariants par symétrie O(3), Comptes-Rendus de l’Académie des Sciences de Paris, série I, 297, 469–471.

    MathSciNet  MATH  Google Scholar 

  • Dos Reis G.L [1984]. Structural stability of equivariant vector fields on two manifolds, Trans. Amer. Math. Soc., 283, 633–643.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Friedrich R, Haken H [1986]. Static, wavelike and chaotic convection in spherical geometries, Phys. Rev. A, 34, 3, 2100–2120.

    CrossRef  Google Scholar 

  • Golubitsky M, Schaeffer D [1982]. Bifurcation with O(3) symmetry including applications to the spherical Bénard problem, Comm. Pure Appl. Math., 35–81.

    Google Scholar 

  • Golubitsky M, Stewart I, Schaeffer D [1988]. Singularities and groups in Bifurcation theory vol. II, Appl. Math. Sci. Ser. 69, Springer Verlag, New-York.

    CrossRef  MATH  Google Scholar 

  • Guckenheimer J, Holmes P [1988]. Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. Soc., 103, 183–192.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Melbourne I, Chossat P, Golubitsky M [1988]. Heteroclinic cycles involving periodic solutions in mode interaction with O(2) symmetry, Preprint, University of Houston.

    Google Scholar 

  • Moutrane E [1988]. Intéraction de modes sphériques dans le problème de Bénard entre deux sphères, thesis, Université de Nice.

    Google Scholar 

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© 1991 Springer-Verlag

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Chossat, P., Armbruster, D. (1991). Structurally stable heteroclinic cycles in a system with O(3) symmetry. In: Roberts, M., Stewart, I. (eds) Singularity Theory and its Applications. Lecture Notes in Mathematics, vol 1463. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085425

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  • DOI: https://doi.org/10.1007/BFb0085425

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53736-6

  • Online ISBN: 978-3-540-47047-2

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