Visualization of Complex ODE Solutions

  • Laurent Testard


This paper presents a visualization method for Complex Ordinary Differential Equations (CODEs) based on real extended phase portraits. This technique enables to visualize functions of classical ODE theory and supports the visual interpretation of numerical integration results. The paper comprises three parts: firstly a presentation of the problem and related existing methods, secondly an introduction to extended phase portraits and their application to complex equations, and thirdly two applications of this visualization technique.


Riemann Surface Phase Portrait Integration Path Visualization Technique Global Error 
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  1. 1.
    R. Aid, Intégration numérique d’équations différentielles ordinaires complexes sans spécification a priori du chemin. Technical report, LMC-IMAG, Grenoble, submitted 1997.Google Scholar
  2. 2.
    R. Aid, L. Levacher, Numerical investigations on global error estimation for ordinary differential equations. Journal on Computational and Applied Mathematics, 1997.Google Scholar
  3. 3.
    C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, 1978.Google Scholar
  4. 4.
    H. Cartan, Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes. Hermann, 1961.Google Scholar
  5. 5.
    M. Fedoriouk, Méthodes asymptotiques pour les équations différentielles ordinaires linéaires. Mir, 1987.Google Scholar
  6. 6.
    E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations I Nonstiff Problems. Springer-Verlag, 1980.Google Scholar
  7. 7.
    J. H. Hubbard, B.H. West, Differential Equations: A Dynamical Systems Approach. Springer-Verlag, 1995.Google Scholar
  8. 8.
    F. Richard-Jung, Le phénomène de Stokes en image - RT 65. Technical report, LMC-Imag, 1991.Google Scholar
  9. 9.
    G. Springer, Introduction to Riemann Surfaces. Addison-Wesley, 1957.Google Scholar
  10. 10.
    D. Stalling, M. Zöckler, H.C. Hege, Fast display of illuminated field lines. IEEE Transactions on Visualization and Computer Graphics, 3: 2, April 1997, pp. 118–128.CrossRefGoogle Scholar
  11. 11.
    L. Testard, Visualisation et calculs en nombres complexes. PhD thesis, INPG, 1997 (submitted).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Laurent Testard
    • 1
  1. 1.Tour des MathsLMC-IMAGGrenobleFrance

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