Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 112
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Afraimovich, Valentin S Moses, Gregory and Young, Todd 2016. Two-dimensional heteroclinic attractor in the generalized Lotka–Volterra system. Nonlinearity, Vol. 29, Issue. 5, p. 1645.

    Ahmed, Hafiz Ushirobira, Rosane Efimov, Denis and Perruquetti, Wilfrid 2016. Robust Synchronization for Multistable Systems. IEEE Transactions on Automatic Control, Vol. 61, Issue. 6, p. 1625.

    Chossat, Pascal and Krupa, Maciej 2016. Heteroclinic Cycles in Hopfield Networks. Journal of Nonlinear Science, Vol. 26, Issue. 2, p. 315.

    Efimov, Denis Schiffer, Johannes and Ortega, Romeo 2016. Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids. International Journal of Control, Vol. 89, Issue. 5, p. 909.

    Guerra, M. Efimov, D. Zheng, G. and Perruquetti, W. 2016. Avoiding local minima in the potential field method using input-to-state stability. Control Engineering Practice, Vol. 55, p. 174.

    Ahmed, Hafiz Ushirobira, Rosane Efimov, Denis and Perruquetti, Wilfrid 2015. 2015 European Control Conference (ECC). p. 181.

    Angeli, David and Efimov, Denis 2015. Characterizations of Input-to-State Stability for Systems With Multiple Invariant Sets. IEEE Transactions on Automatic Control, Vol. 60, Issue. 12, p. 3242.

    Chen, Xiaojing Jiang, Jifa and Niu, Lei 2015. On Lotka--Volterra Equations with Identical Minimal Intrinsic Growth Rate. SIAM Journal on Applied Dynamical Systems, Vol. 14, Issue. 3, p. 1558.

    Efimov, Denis Ortega, Romeo and Schiffer, Johannes 2015. 2015 American Control Conference (ACC). p. 4664.

    Grines, Evgeny A. and Osipov, Grigory V. 2015. On constructing simple examples of three-dimensional flows with multiple heteroclinic cycles. Regular and Chaotic Dynamics, Vol. 20, Issue. 6, p. 679.

    Horchler, Andrew D Daltorio, Kathryn A Chiel, Hillel J and Quinn, Roger D 2015. Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control. Bioinspiration & Biomimetics, Vol. 10, Issue. 2, p. 026001.

    Korotkov, Alexander G. Kazakov, Alexey O. and Osipov, Grigory V. 2015. Sequential dynamics in the motif of excitatory coupled elements. Regular and Chaotic Dynamics, Vol. 20, Issue. 6, p. 701.

    Palacios, Antonio In, Visarath and Longhini, Patrick 2015. Symmetry-Breaking as a Paradigm to Design Highly-Sensitive Sensor Systems. Symmetry, Vol. 7, Issue. 2, p. 1122.

    Shaw, Kendrick M. Lyttle, David N. Gill, Jeffrey P. Cullins, Miranda J. McManus, Jeffrey M. Lu, Hui Thomas, Peter J. and Chiel, Hillel J. 2015. The significance of dynamical architecture for adaptive responses to mechanical loads during rhythmic behavior. Journal of Computational Neuroscience, Vol. 38, Issue. 1, p. 25.

    Curbelo, Jezabel and Mancho, Ana M. 2014. Symmetry and plate-like convection in fluids with temperature-dependent viscosity. Physics of Fluids, Vol. 26, Issue. 1, p. 016602.

    Curbelo, J. and Mancho, A.M. 2014. Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature. Communications in Nonlinear Science and Numerical Simulation, Vol. 19, Issue. 3, p. 538.

    Levanova, T.A. Osipov, G.V. and Pikovsky, A. 2014. Coherence properties of cycling chaos. Communications in Nonlinear Science and Numerical Simulation, Vol. 19, Issue. 8, p. 2734.

    Angeli, David and Efimov, Denis 2013. 52nd IEEE Conference on Decision and Control. p. 5897.

    Ashwin, Peter and Postlethwaite, Claire 2013. On designing heteroclinic networks from graphs. Physica D: Nonlinear Phenomena, Vol. 265, p. 26.

    Curbelo, J. and Mancho, A. M. 2013. Bifurcations and dynamics in convection with temperature-dependent viscosity in the presence of the O(2) symmetry. Physical Review E, Vol. 88, Issue. 4,

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 103, Issue 1
  • January 1988, pp. 189-192

Structurally stable heteroclinic cycles

  • John Guckenheimer (a1) and Philip Holmes (a1)
  • DOI:
  • Published online: 24 October 2008

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields equivariant with respect to a symmetry group. In the space X(M) of Cr vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space XG(M) ⊂ X(M) of vector fields equivariant with respect to a symmetry group G, the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ3 equivariant with respect to a particular finite subgroup GO(3) such that each XU has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3]F. H. Busse and K. E. Heikes . Convection in a rotating layer: a simple case of turbulence. Science 208 (1980), 173175.

[4]J. Guckenheimer and P. Holmes . Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, 1983).

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *