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On a codimension two bifurcation

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Dynamical Systems and Turbulence, Warwick 1980

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 898))

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References

  1. V.I. Arnold, Lectures on Bifurcations in Versal Families, Russian Math. Surveys 27 (1972) 54–123.

    Article  ADS  Google Scholar 

  2. J.F.G. Auchmuty, Bifurcating waves, Annals NY Acad. Sci. 316 (1979) 263–278.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. J.F.G. Auchmuty & G. Nicolis, Bifurcation analysis of non-linear reaction-diffusion equations I, Bull. Math. Biology 37 (1975), 323–365.

    MathSciNet  MATH  Google Scholar 

  4. J.F.G. Auchmuty & G. Nicolis, Bifurcation analysis of reaction-diffusion equations III, Chemical Oscillations, Bull. Math. Biology 38, (1976), 325–249.

    MathSciNet  MATH  Google Scholar 

  5. R.I. Bogdanov, Orbital equivalence of singular points of vector fields on the plane, Funct. Anal. Appl. 10, 316–317, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  6. R.I. Bogdanov, Versal deformations of a singular point of a vector field on a plane in the case of zero eigenvalues, Proceedings of the I.G. Petrovskii Seminar, 2, 37–65, 1976.

    MathSciNet  Google Scholar 

  7. H. Degn, L. Olsen & J. Perram, Bistability, Oscillation, and Chaos in an enzyme reaction, Annals N.Y. Acad. Sci. 316, (1979) 623–637.

    Article  ADS  Google Scholar 

  8. F. Dumortier, Singularities of vector fields in the plane, Journal of Differential Equations, 23, 53–106, 1977.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. N.K. Gavrilov & L.P. Silnikov, On three dimensional dynamical systems close to systems with a structurally unstable homoclinic curve, Math. USSR, Sb. 17, 467–485, 1972, and 19, 139–156, 1973.

    Article  MathSciNet  Google Scholar 

  10. M. Golubitsky & D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32 (1979) 21–98.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Herschkowitz-Kaufman, Bifurcation analysis of reaction-diffusion equations II, Bull. Math. Biology 37 (1975) 589–636.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Herschkowitz-Kaufman & T. Erneux, The bifurcation diagram of model chemical reactions, Annals N.Y. Acad Sci. 316 (1979) 296–313.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. P. Holmes & J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica 14 (1978) 367–384.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. J.P. Keener, Secondary bifurcation in non-linear diffusion reaction equations II, Studies in Appl. Math. 55 (1976), 187–211.

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Kopell & L. Howard, Target patterns and horseshoes from a perturbed central force problem: some temporally periodic solutions to reaction diffusion equations. Preprint 1979.

    Google Scholar 

  16. R. Lefever & I. Prigogine, Symmetry breaking instabilities in dissipative systems II, J. Chem. Phys. 48 (1968), 1695–1700.

    Article  ADS  Google Scholar 

  17. T. Mahar & B. Matkowsky, A model chemical reaction exhibiting secondary bifurcation, SIAM J. Appl. Math. 32 (1977), 394–404.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Marsden, Qualitative methods in bifurcation theory, Bull. Am. Math. Soc. 84 (1978) 1125–1148.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Marsden & M. McCracken, The Hopf Bifurcation Theorem and Its Applications, Springer Verlag, 1976.

    Google Scholar 

  20. S. Newhouse, Diffeomorphisms with Infinitely Many Sinks, Topology, 12, 1974, 9–18.

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, mimeographed, IHES, 1977.

    Google Scholar 

  22. S. Newhouse, J. Palis, & F. Takens, Stable Arcs of Diffeomorphisms, Bull. Am. Math. Soc. 82, 499–502, 1976, and to appear.

    Article  MathSciNet  MATH  Google Scholar 

  23. L.F. Olsen & H. Degn, Chaos in an enzyme reaction, Nature, 267, 1977, 177–178.

    Article  ADS  Google Scholar 

  24. J. Palis & F. Takens, Topological equivalence of normally hyperbolic dynamical systems, Topology, 16, 335–345, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Ruelle, Sensitive dependence on initial condition and turbulent behaviour of dynamical systems, conference on Bifurcation theory and its Applications, N.Y. Academy of Sciences, 1977.

    Google Scholar 

  26. D. Schaeffer & M. Golubitsky, Bifurcation analysis near a double eigenvalue of a model chemical reaction MRC Technical Report, #1859, 1978.

    Google Scholar 

  27. R.A. Schmitz, K.R. Graziani & J.L. Hudson, Experimental evidence of chaotic states in the Belousov-Zhabotinskii reaction, J. Chem. Phys. 67, 1977, 3040–3044.

    Article  ADS  Google Scholar 

  28. R. Schmitz, G. Renola & P. Garrigan, Observations of complex dynamic behaviour in the H2−O2 reaction on nickel. Annals N.Y. Acad, Sci. 316 (1979), 638–651.

    Article  ADS  Google Scholar 

  29. L.P. Sil'nikov, A contribution to the problem of the structure of an extended neighbourhood of a structurally stable equilibrium of saddle-focus type, Math. USSR Sb. 10, 91–102, 1970.

    Article  Google Scholar 

  30. S. Smale, Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology, Princeton, 1965, 63–80.

    Google Scholar 

  31. P. Grace Sørensen, Experimental investigations of behaviour and stability properties of attractors corresponding to burst phenomena in the open Belousov reaction, Annals N.Y. Acad. Sci. 316 (1979) 667–675.

    Article  ADS  MATH  Google Scholar 

  32. J. Sotomayor, Bifurcations of vector fields on two dimensional manifolds, Publ. IHES, #43, 1–46, 1973.

    Google Scholar 

  33. J. Sotomayor, Generic bifurcations of dynamical systems, Dynamical Systems, ed. M. Peixoto, Academic Press, 1973, 561–582.

    Google Scholar 

  34. F. Takens, Singularities of Vector Fields, Publ., IHES, #43, 47–100, 1973.

    MathSciNet  MATH  Google Scholar 

  35. F. Takens, Forced Oscillations and Bifurcations, Applications of Global Analysis, Communications of Maths. Institute, Rijksuniversiteit, Utrecht, 3, 1974, 1–59.

    MathSciNet  MATH  Google Scholar 

  36. R. Thom, Structural Stability and Morphogenesis, W.A. Benjamin Inc., Reading, Mass. 1975.

    MATH  Google Scholar 

  37. A. Andronov, E. Leontovich, I. Gordon & A. Maier, The Theory of Bifurcation of Plane Dynamical Systems, 1971.

    Google Scholar 

  38. J. Gollub & H. Swinney, Onset of Turbulence in a Rotating Fluid, Physical Review Letters, 35, 1975, 927–930.

    Article  ADS  Google Scholar 

  39. M. Herman, Sur la Conjugasion Differentiable des Diffeomorphismes du Cercle a des Rotations, Publ. IHES, 49, 1979, 5–234.

    Article  MathSciNet  MATH  Google Scholar 

  40. Y. Kuramoto, Diffusion Induced Chaos in Reacting Systems, Suppl. Prog. Theo. Phys. 64, 1978, 346–367.

    Article  ADS  Google Scholar 

  41. W. Langford, Periodic and Steady-State Mode Interactions Lead to Tori, SIAM J. Appl. Math., 37, 1979, 22–48.

    Article  MathSciNet  MATH  Google Scholar 

  42. D. Ruelle & F. Takens, On the Nature of Turbulence, Comm. Math. Phys., 20, 1971, 167–192.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. V.I. Arnold, Small denominators I, Mappings of the circle onto itself, Izv. Akad. Nauk, SSSR Ser. Mat. 25, 1961, 21–86.

    MathSciNet  Google Scholar 

  44. P. Holmes & J. Marsden, A Partial Differential Equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, preprint.

    Google Scholar 

  45. P. Holmes & F. Moon, A Magnetoelastic Strange Attractor, J. Sound and Vibration 65, 1979, 275–296.

    Article  ADS  MATH  Google Scholar 

  46. H.W. Broer, Bifurcations of singularities in volume preserving vector fields, thesis, Rijksuniversiteit te Groningen, 1979.

    Google Scholar 

  47. W.F. Langford & G Iooss, Interactions of Hopf and pitchford bifurcations, Workshop on Bifurcation Problems, Birkhauser Lecture Notes, 1980.

    Google Scholar 

  48. P. Holmes, A strange family of three dimensional vector fields near a degenerate singularity, J. Diff. Eq. 37, 382–403, 1980.

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Cruz, California, USA

    John Guckenheimer

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  1. John Guckenheimer
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David Rand Lai-Sang Young

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© 1981 Springer-Verlag

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Guckenheimer, J. (1981). On a codimension two bifurcation. In: Rand, D., Young, LS. (eds) Dynamical Systems and Turbulence, Warwick 1980. Lecture Notes in Mathematics, vol 898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091910

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  • DOI: https://doi.org/10.1007/BFb0091910

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  • Print ISBN: 978-3-540-11171-9

  • Online ISBN: 978-3-540-38945-3

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