An algorithm for symbolic computation of center manifolds

  • Emilio Freire
  • Estanislao Gamero
  • Enrique Ponce
  • Leopoldo G. Franquelo
Differential Equations
Part of the Lecture Notes in Computer Science book series (LNCS, volume 358)

Abstract

A useful technique for the study of local bifurcations is the center manifold theory because a dimensional reduction is achieved. The computation of Taylor series approximations of center manifolds gives rise to several difficulties regarding the operational complexity and the computational effort. Previous works proceed in such a way that the computational effort is not optimized. In this paper an algorithm for center manifolds well suited to symbolic computation is presented. The algorithm is organized according to an iterative scheme making good use of the previous steps, thereby minimizing the number of operations. The results of two examples obtained through a REDUCE 3.2 implementation of the algorithm are included.

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References

  1. 1.
    J. Aracil, E. Freire and E. Ponce: Order through fluctuations, and systems dynamics models, Environment and Planning B: Planning and Design, vol. 12, pp 103–112 (1985)Google Scholar
  2. 2.
    J. Carr: Applications of Centre Manifold Theory, Appl. Math. Sci. Series vol 35, Springer, New York (1981).Google Scholar
  3. 3.
    S. Chow and J. K. Hale: Methods of Bifurcation Theory. Springer, New York (1982).Google Scholar
  4. 4.
    E. Freire and E. Gamero: Computación simbólica de variedades de centros con aplicación al estudio de bifurcaciones locales de codimensión dos en un sistema electrónico autónomo, in ACTAS X CEDYA, Valencia (1987).Google Scholar
  5. 5.
    E. Freire and E. Ponce: Bifurcaciones estáticas en un modelo de interacción de dos sistemas con crecimiento limitado, in ACTAS X CEDYA, Valencia (1987).Google Scholar
  6. 6.
    E. Freire, L.G. Franquelo and J. Aracil: Periodicity and Chaos in an Autonomous Electronic System, IEEE Transactions on CAS, vol 31, pp. 237–247 (1984).Google Scholar
  7. 7.
    A.C. Hearn (ed.): REDUCE 3.2, The Rand Corporation, Santa Monica (1985).Google Scholar
  8. 8.
    K. R. Meyer and D. S. Schmidt: Entrainment Domains, Funkcialaj Ekvacioj, vol 20, pp 171–192 (1977).Google Scholar
  9. 9.
    E. Ponce: Técnicas de análisis cualitativo en sistemas dinámicos, Ph. D. Thesis, Universidad de Sevilla (1987).Google Scholar
  10. 10.
    R. H. Rand and W. L. Keith: Normal Form and Center Manifold Calculations on MACSYMA, in Applications of Computer Algebra, R. Pavelle (ed.), Kluwer Academic Publishers, Boston (1985).Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Emilio Freire
    • 1
  • Estanislao Gamero
    • 1
  • Enrique Ponce
    • 1
  • Leopoldo G. Franquelo
    • 2
  1. 1.Department of Applied MathematicsEscuela Superior Ingenieros IndustrialesSevillaSpain
  2. 2.Department of Electronic Engineering, Systems and AutomaticsEscuela Superior Ingenieros IndustrialesSevillaSpain

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