Japan Journal of Applied Mathematics

, Volume 4, Issue 3, pp 393–431 | Cite as

Constrained systems, characteristic surfaces, and normal forms

  • Hiroe Oka


ODEs with a small parameter ε multiplying derivatives, which have been studied as a singular perturbation problem, are formulated in a coordinate-free manner, as a pair of a vector field and a tensor field, in order to treat them as a bifurcation problem. This formulation is an extension of the classical interpretation of autonomous ODEs as vector fields, and it is called a constrained system. A method to obtain normal forms for constrained systems is given as well as several results of computing them. Unfoldings of these normal forms include a large part of typical behaviors observed in such ODEs with ε. The local classification of characteristic surfaces is also discussed.

Key words

constrained system characteristic surface normal form singular perturbation bifurcation 


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  1. [1]
    V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations. Springer-Verlag. New York, 1983.Google Scholar
  2. [2]
    R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications. Addison-Wesley, London, 1983.MATHGoogle Scholar
  3. [3]
    E. Benoit, Canards deR 3. Thesis, Nice, 1983.Google Scholar
  4. [4]
    Th. Bröcker, Differentiable Germs and Catastrophes. London Math. Soc. Lecture Note Ser. 17, Cambridge Univ. Press, 1975.Google Scholar
  5. [5]
    E. Benoit, J. L. Callot, F. Diener and M. Diener, Chasse au canard. Collect. Math.,32 (1980), 37–119.Google Scholar
  6. [6]
    L. O. Chua and H. Kokubu, Normal forms for nonlinear vector fields: I. In preparation.Google Scholar
  7. [7]
    L. O. Chua and H. Oka, Normal forms for constrained, nonlinear equations. In preparation.Google Scholar
  8. [8]
    M. Diener, The canard unchained or how fast/slow dynamical systems bifurcate. Math. Intelligencer,6 (1984), 38–49.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    N. Fenichel, Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations,31 (1979), 53–98.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    N. Fenichel, Oscillatory bifurcations in singular perturbation theory, II. Fast oscillations. SIAM J. Math. Anal.,14 (1983), 868–874.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities GTM 14, Springer-Verlag, New York, 1973.Google Scholar
  12. [12]
    J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Appl. Math. Sci. 42, Springer-Verlag, New York, 1983.Google Scholar
  13. [13]
    W. Greub, S. Halperin and R. Vanstone, Connections, Curvature, and Cohomology vol. 1. Academic Press, New York, 1972.MATHGoogle Scholar
  14. [14]
    M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory. vol. 1. Appl. Math. Sci. 51, Springer-Verlag, New York, 1985.Google Scholar
  15. [15]
    G. Ikegami, Vector fields tangent to foliations. Japan. J. Math.,12 (1986), 95–120.MATHMathSciNetGoogle Scholar
  16. [16]
    G. Ikegami, Singular perturbations in foliations, Preprint.Google Scholar
  17. [17]
    E. N. Lorenz, Deterministic nonperiodic flow. J. Atom. Sci.,20 (1963), 130–141.CrossRefGoogle Scholar
  18. [18]
    E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations. Plenum Press, New York, 1980.MATHGoogle Scholar
  19. [19]
    H. Oka and H. Kokubu, An, approach to constrained equations and strange attractors. Patterns and Waves—Qualitative Analysis of Nonlinear Differential Equations—(eds. T. Nishida, M. Mimura and H. Fujii), Kinokuniya and North-Holland, 1986, 607–630.Google Scholar
  20. [20]
    H. Oka and H. Kokubu, Constrained Lorenz-like attractors, Japan J. Appl. Math.,2 (1985), 495–500.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    F. Takens, Singularities of vector fields. Publ. Math. IHES,43 (1974), 47–100.MathSciNetGoogle Scholar
  22. [22]
    F. Takens, Constrained equations: a study of implicit differential equations and their discontinuous solutions. Lecture Notes in Math., 525, 1976, 143–234.CrossRefGoogle Scholar
  23. [23]
    F. Takens, Implicit differential equations: some open problems. Lecture Notes in Math., 535, 1976, 237–253.CrossRefMathSciNetGoogle Scholar
  24. [24]
    F. Takens, Transitions from periodic to strange attractors in constrained equations. Preprint.Google Scholar
  25. [25]
    S. Ushiki, Normal forms for singularities of vector fields. Japan J. Appl. Math.,1 (1984), 1–37.MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    S. Ushiki and R. Lozi, Confinor and anti-confinor in constrained “Lorenz” system. Japan J. Appl. Math.,4 (1987), 433–454.MATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    E. C. Zeeman, Differential equations for the heartbeat and nerve impulse. Dynamical Systems, Salvador 1971. Academic Press, New York, 1973, 683–741.Google Scholar
  28. [28]
    E. C. Zeeman, Levels of structure in catastrophe theory illustrated by applications in the social and biological sciences. Proceedings of The International Congress of Mathematicians, Vancouver, 1974, vol. 2, 533–546.Google Scholar
  29. [29]
    A. K. Zvonkin and M. A. Shubin, Non-standard analysis and singular perturbations of ordinary differential equations. Russian Math. Surveys,39, (1984), 77–127.CrossRefMathSciNetGoogle Scholar

Copyright information

© JJAM Publishing Committee 1987

Authors and Affiliations

  • Hiroe Oka
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan

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