Heteroclinic Cycles in Symmetrically Coupled Systems

  • Michael Field
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 115)


A characteristic feature of symmetric dynamics is the presence of robust heteroclinic cycles. Although this phenomenon was first described by dos Reis in 1978 (see [25]), it only gained wide attention in the dynamics community after the work of Guckenheimer and Holmes [18] on a dynamical system used by Busse and Clever [7] as a model of fluid convection. Robust heteroclinic cycles also occur in models of population dynamics, see [20, 21]. The existence of robust cycles in equivariant dynamics can be viewed as a special instance of the fact that in equivariant dynamical systems intersections of invariant manifolds can be stable under perturbation even though intersections are not transverse [10, 12]. Indeed, this failure of transversality is a prerequisite for a cycle between hyperbolic equilibria since transversality between stable and unstable manifolds of hyperbolic equilibria implies no cycles.


Wreath Product Internal Symmetry Heteroclinic Cycle Transitive Subgroup Hyperbolic Equilibrium 
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  1. [1]
    P ASHWIN, Chaotic intermittency of patterns in symmetric systems, these proceedings.Google Scholar
  2. [2]
    P ASHWIN AND M J FIELD, Heteroclinic networks in coupled cell systems, to appear in Arch. Rat. Mech. and Anal.Google Scholar
  3. [3]
    P ASHWIN AND A M RUCKLIDGE, Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection, to appear in Physica D, (1998).Google Scholar
  4. [4]
    A BACK, J GUCKENHEIMER, M MYERS, F WICKLIN, AND P WORFOLK, dstool: Computer Assisted Exploration of Dynamical Systems, Notices AMS, 39(4), (1992), 303–309.Google Scholar
  5. [5]
    E BIERSTONE, General position of equivariant maps, Trans. Amer. Math. Soc., 234, (1977), 447–466.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    E BIERSTONE, The structure of orbit spaces and the singularities of equivariant mappings, Instituto de Matemática Oura e Aplicada, Rio de Janeiro, 1980.zbMATHGoogle Scholar
  7. [7]
    F H BUSSE AND R M CLEVER, Nonstationary Convection in a Rotating System, in Recent Developments in Theoretical and Experimental Fluid Mechanics, ed. U. Müller, K.G. Rösner and B. Schmidt, Springer, Berlin, (1979), 376–385.CrossRefGoogle Scholar
  8. [8]
    M DELLNITZ, M FIELD, M GOLUBITSKY, A HOHMANN AND J MA, Cycling Chaos, Intern. J. Bifur. & Chaos, 5(4), (1995), 1243–1247.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    B DIONNE, M GOLUBITSKY AND I STEWART, Coupled cells with internal symmetry: I. Wreath products, Nonlinearity, 9, (1996), 559–574.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    M J FIELD, Equivariant dynamical systems, Bull. Amer. Math. Soc., (1970), 1314–1318.Google Scholar
  11. [11]
    M J FIELD, Transversality in G-manifolds, Trans. Amer. Math. Soc., 231, (1977), 429–450.MathSciNetzbMATHGoogle Scholar
  12. [12]
    M J FIELD, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259(1), (1980), 185–205.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    M J FIELD AND R W RICHARDSON, Symmetry breaking and branching patterns in equivariant bifurcation theory II, Arch. Rational Mech. Anal., 120, (1992), 147–190.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    M J FIELD AND J W SWIFT, Static bifurcation to limit cycles and heteroclinic cycles, Nonlinearity, 4(4), (1991), 1001–1043.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    M J FIELD, Dynamics, Bifurcation and Symmetry, Pitman Research Notes in Mathematics, 356, (1996).Google Scholar
  16. [16]
    D B Gillis. Coupled cell systems and symmetry, thesis, University of Houston, (1996).Google Scholar
  17. [17]
    M GOLUBITSKY, D G SCHAEFFER AND I N STEWART, Singularities and Groups in Bifurcation Theory, Vol. II, Appl. Math. Sci., 69, Springer-Verlag, New York, (1988).Google Scholar
  18. [18]
    J GUCKENHEIMER AND P HOLMES, Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. Soc., 103, (1988), 189–192.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    J GUCKENHEIMER AND P WORFOLK, Instant chaos, Nonlinearity, 5, (1992), 1211–1222.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    J HOFBAUER, Heteroclinic cycles on the simplex, Proc Int. Conf. Nonlinear Oscillations, Janos Bolyai Math. Soc Budapest, (1987).Google Scholar
  21. [21]
    J HOFBAUER AND K SIGMUND, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, (1988).zbMATHGoogle Scholar
  22. [22]
    M KRUPA, Robust heteroclinic cycles, Journal of Nonlinear Science, 7, (1997), 129–176.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    M KRUPA AND I MELBOURNE, Asymptotic stability of heteroclinic cycles in systems with symmetry, Erg. Th. Dyn. Sys., 15, (1995), 121–147.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    I MELBOURNE, P CHOSSAT AND M GOLUBITSKY, Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry, Proc. Royal Soc. Edinburgh, 113A, (1989), 315–345.MathSciNetCrossRefGoogle Scholar
  25. [25]
    G L DOS REIS, Structural stability of equivariant vector fields on two-manifolds, Trans. Amer. Math. Soc., 283, (1984), 633–642.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Michael Field
    • 1
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA

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