Chaos and the Effect of Noise for the Double Hopf Bifurcation with 2:1 Resonance

  • Michael R. E. Proctor
  • David W. Hughes
Part of the NATO ASI Series book series (NSSB, volume 225)


A major goal of bifurcation theory is the identification and classification of important degenerate bifurcations of low codimension. Although many have been extensively studied (for a non-exhaustive list see Guckenheimer & Holmes 1983) key exceptions include degenerate Hopf or oscillatory bifurcations with strong resonance. In this paper we consider the problem of 2:1 resonance, so that at the bifurcation point there are considered to be two neutrally stable infinitesimal oscillatory solutions with frequencies in the ratio 2:1. The normal form for this problem was discussed by Knobloch & Proctor (1988) who noted that it contains quadratic as well as the usual non-resonant cubic terms, leading to a system of three coupled real equations, unlike the non-resonant case where the phase equation decouples from the other two. Knobloch & Proctor investigated the normal form including all terms up to cubic order — they found many of the principal transitions, but did not discover any régime in which chaos occurred, and did not identify regions in the unfolding diagram where the dynamics was controlled by the linear and quadratic terms alone. In the present paper we rectify this deficiency by showing that when the linear growth rates γ1,2 of the modes of normalised frequency 1,2 satisfy \( ( - \gamma _1 ) \gg \gamma _2 > 0 \) there are indeed bounded solutions when the normal form is truncated at quadratic order. Furthermore, in this régime the governing o.d.e.’s may be replaced by a one-dimensional map, which can be calculated analytically in certain special cases.


Normal Form Bifurcation Diagram Additive Noise Slow Phase Fast Phase 
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  1. Guckenheimer, J. Sz Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag.Google Scholar
  2. Hughes, D.W. & Proctor, M.R.E. 1990a A low-order model for the shear instability of convection: chaos and the effect of noise. Nonlinearity 3.Google Scholar
  3. Hughes, D.W. & Proctor, M.R.E. 1990b Chaos and the effect of noise in a model of three-wave mode coupling (in preparation). Google Scholar
  4. Hughes, D.W. & Proctor, M.R.E. 1990c Chaos and the effect of noise in the double Hopf bifurcation with 2:1 resonance (in preparation). Google Scholar
  5. Knobloch, E. & Proctor, M.R.E. 1988 The double Hopf bifurcation with 2:1 resonance. Proc. R. Soc. Lond. A415, 61–90.MathSciNetADSGoogle Scholar
  6. Mackay, R.S. & Tresser, C. 1987 Some flesh on the skeleton: the bifurcation structure of bimodal maps. Physica 27D, 412–422.MathSciNetADSGoogle Scholar
  7. Vyshkind, S.Ya. & Rabinovich, M.I. 1976 The phase stochastization mechanism and the structure of wave turbulence in dissipative media. Sov. Phys. JETP 44, 292–299.ADSGoogle Scholar
  8. Wersinger, J.-M., Finn, J.M. & Ott, E. 1980 Bifurcation and “strange” behavior in instability saturation by nonlinear three-wave mode coupling. Phys. Fluids 23, 1142–1154.MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Michael R. E. Proctor
    • 1
  • David W. Hughes
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeEngland

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