New Numerical Methods for High Dimensional Hopf Bifurcation Problems

  • J. Wu
  • K. Zhou
Part of the Springer Series in Nonlinear Dynamics book series (SSNONLINEAR)

Abstract

In the paper a new numerical method for the analysis of both static and Hopf bifurcation is presented. The study of the problem of solution bifurcation is reduced to the study of the local behavior near the singular points of the vector field. In the case of the Hopf bifurcation, the problem is reduced to searching for a pair of complex conjugate eigenvalues with maximum norm of a matrix. Two examples support the introduced theory.

Keywords

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H.J. Wacker(ed.): Continuation methods. Academic Press, New York 1978MATHGoogle Scholar
  2. 2.
    E. Wasserstrom: Numerical solutions by the continuation method. SIAM Rev. 15, 89–119 (1973)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    R. Menzel: Numerical determination of multiple bifurcation points. In [1], 310–318Google Scholar
  4. 4.
    G Moore, A. Spence: The calculation of turning points of nonlinear equations. SIAM J. Numer. Anal. 17, 567–576 (1980)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    HB. Keller: Numerical solution of bifurcation and nonlinear eigenvalue problems. In applications of bifurcation theory. Edited by P.H. Rabinowitz, Academic Press, Inc., 1977Google Scholar
  6. 6.
    Ji Hai-po, Wu Ji-ke, Hu Hai-chang: Geometric description and computational methods for bifurcation. Science in China(A) 9, 597–607 (1990)Google Scholar
  7. 7.
    Su Xian-yue: The nonlinear analysis of rotationary shells. Ph. D. Thesis, Peking University (1985)Google Scholar
  8. 8.
    Su Xian-yue, Wang Ying-jian, Wu Ji-ke, Hu Hai-chang: A numerical method for the nonlinear systems with parameters. Computational Structural Mechanics and Applications 3, 1–9 (1988)Google Scholar
  9. 9.
    Wu Ji-ke, Teng Ning-jun and Yuang Yong: Bifurcation problems and their numerical methods. Mechanics and Practice 4, 1–7 (1987)Google Scholar
  10. 10.
    A.A. Andronov and A. Witt: Sur la théorie mathématiques des autooscillations. C.R.A-cad. Sci. Paris 190, 256–258 (1930)Google Scholar
  11. 11.
    B.D. Hassard, N.D. Kazarinoff and Y. H. Wan: Theory and applications of Hopf bifurcation. Cambridge University Press 1981MATHGoogle Scholar
  12. 12.
    E. Hopf: Abzweigung einer periodischen Lösung von einer stationären Lösung eines differential Systems. Ber. Math-Phys. Sachsische Adademie der Wissenschaften Leipzig 94, 1–22 (1942)Google Scholar
  13. 13.
    H. Poincaré: Les méthodes nouvelles de la mécanique céleste. Vol.I Paris (1892)Google Scholar
  14. 14.
    A. Kelley: The stable, center-stable, center, center-unstable and unstable manifolds. J. Diff. Eqns. 3, 546–570 (1967)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    I.E. Marsden and M. McCracken: The Hopf bifurcation and its applications. Springer-Verlag, New York 1976MATHCrossRefGoogle Scholar
  16. 16.
    Wu Ji-ke and Zhou Kun: Numerical computation for high dimension Hopf bifurcation. Acta Scientiarum Naturalium Universitatis Pekinesis (to appear)Google Scholar
  17. 17.
    W. Hahn: Theory and application of Lyapunov’s direct method. Prentice-Hall 1963Google Scholar
  18. 18.
    S.N. Chow and J.K. Hale: Methods of bifurcation theory. Springer-Verlag, New York 1982MATHCrossRefGoogle Scholar
  19. 19.
    Wu Ji-ke and Shao Xiu-ming: The circulant matrix and its applications in the computation of structures. Mathematicae Numericae Sinica 2, 144–153 (1979)Google Scholar
  20. 20.
    C. Sparrow: The Lorenz equations: bifurcations, chaos, and strange attractors. Springer-Verlag, New York 1982MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • J. Wu
    • 1
  • K. Zhou
    • 1
  1. 1.Department of MechanicsPeking UniversityBeijingPRC

Personalised recommendations