The Numerical Detection of Hopf Bifurcation Points

  • G. Moore
  • T. J. Garratt
  • A. Spence
Part of the NATO ASI Series book series (ASIC, volume 313)

Abstract

During the computation of a one-parameter set of steady solutions to a time dependent problem, it is often of great interest to know when periodic behaviour can occur. This Hopf bifurcation is characterised by the linearisation about a steady state solution possessing a purely imaginary pair of eigenvalues. The cheap and reliable detection of such bifurcation is not however straightforward. The present paper examines the problem and suggests various strategies for solving it.

Keywords

Hopf Bifurcation Invariant Subspace Krylov Subspace Negative Real Part Positive Real Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    L. V. Ahlfors. Complex Analysis. McGraw Hill, New York, 1966.MATHGoogle Scholar
  2. [2]
    E. J. Doedel, J. P. Kernevez. AUTO: software for continuation and bifurcation problems in ordinary differential equations. Appl. Math. Tech. Rep., Cal. Tech., 1986.Google Scholar
  3. [3]
    P. J. Eberlein. A Jacobi-like method for the automatic computation of eigenvalues and eigenvectors of an abitrary matrix. J. Soc. Ind. Appl. Math., 10, pages 74–88, 1962.MathSciNetCrossRefGoogle Scholar
  4. [4]
    J.N. Franklin. Matrix Theory. Prentice-Hall, New Jersey, 1968.MATHGoogle Scholar
  5. [5]
    F. R. Gantmacher . Matrix Theory Vol. II. Chelsea Publishing Co., New York, 1959.Google Scholar
  6. [6]
    P. Henrici. Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices. Numer. Math., 4, pages 24–40, 1962.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    P. Henrici. Applied and Computational Complex Analysis Vol. 1. Wiley-Inter science, New York, 1974.MATHGoogle Scholar
  8. [8]
    D. Ho, F. Chatelin,, M. Bennani. Arnoldi-Tchebyshev procedure for large scale nonsymmetric matrices. Math. Mod. Num. Anal., 24, pages 53–65, 1990.MathSciNetMATHGoogle Scholar
  9. [9]
    A. S. Householder. The Numerical Treatment of a Single Nonlinear Equation. McGraw Hill, New York, 1970.MATHGoogle Scholar
  10. [10]
    W. Kahan, B.N. Parlett, E. Jiang. Residual bounds on approximate eigensystems of non-normal matrices. SIAM J. Numer. Anal., 19, pages 470–484, 1982.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    P. Lancaster. Theory of Matrices. Academic, New York, 1969.MATHGoogle Scholar
  12. [12]
    T.A. Manteuffel. The Tchebyshev iteration for nonsymmetric linear systems. Numer. Math., 28, pages 307–327, 1977.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    T.A. Manteuffel. Adaptive procedure for estimation of parameter for the non-symmetric Tschebyshev iteration. Numer. Math., 31, pages 183–208, 1978.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    B. Nour-Omid, B.N. Parlett, R. Taylor. Lanczoz versus subspace iteration for the solution of eigenvalue problems. Int. J. Numer. Meth. Eng., pages 859–871, 1983.Google Scholar
  15. [15]
    D. O’Leary, G.W. Stewart, J.S. Vandergraft. Estimating the largest eigenvalue of a positive definite matrix. Math. Comp., 33, pages 1289–1292, 1979.MathSciNetMATHGoogle Scholar
  16. [16]
    B.N. Parlett. The Symmetric Eigenvalue Problem. Prentice Hall, New Jersey, 1980.MATHGoogle Scholar
  17. [17]
    B.N. Parlett. Laguerre’s method applied to the matrix eigenvalue problem. Math. Comp., 18, pages 464–487, 1964.MathSciNetMATHGoogle Scholar
  18. [18]
    B.N. Parlett, H. Simon, L.M. Stringer. On estimating the largest eigenvalue with the Lanczos algorithm. Math. Comp., 33, pages 153–165, 1982.MathSciNetCrossRefGoogle Scholar
  19. [19]
    B.N. Parlett, D.R. Taylor, Z.A. Liu. A look-ahead Lanzcos algorithm for unsym-metric matrices. Math. Comp., 44 pages 105–124, 1985.MathSciNetMATHGoogle Scholar
  20. [20]
    A. Ruhe. On the quadratic convergence of a generalisation of the Jacobi method to arbitrary matrices. BIT, 8, pages 210–231, 1968.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    H. Rutishauser. Solution of eigenvalue problems with the LR transformation. Nat. Bur. Stand. Appl. Math., Ser. 49, pages 47–81, 1958.MathSciNetGoogle Scholar
  22. [22]
    H. Rutishauser. Une methóde pour le calcul des valeurs propres des matrices non- symétriques. Comptes Rendus, 259, page 2758, 1964.MathSciNetMATHGoogle Scholar
  23. [23]
    Y. Saad. Variations of Arnoldi’s method for computing eigenelements of large unsym- metric matrices. Lin. Alg. & Appl., 34, pages 269–295, 1980.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Y. Saad. Least squares polynomials in the complex plane with applications to solving sparse non-symmetric matrix problems. Research Report YALEU/DCS/RR-276, 1983.Google Scholar
  25. [25]
    Y. Saad. Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems. Math. Comp., 42, pages 567–588, 1984.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    G.W. Stewart . Simultaneous iteration for computing invariant subspaces of non- hermitian matrices. Numer. Math 25, pages 123–136, 1976.MATHCrossRefGoogle Scholar
  27. [27]
    F. Tibor. Normal equivalent to an arbitrary diagonalisable matrix. Lin. Alg. & Appl., 51, pages 153–162, 1983.MATHCrossRefGoogle Scholar
  28. [28]
    K. Veselic. On a class of Jacobi-like procedures for diagonalising arbitrary real matrices. Numer. Math., 33, pages 157–172, 1979.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    J. H. Wilkinson. The Algebraic Eigenvalue Problem. Oxford Univ. Press, Oxford, 1965.MATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • G. Moore
    • 1
  • T. J. Garratt
    • 2
  • A. Spence
    • 2
  1. 1.Department of MathematicsImperial CollegeLondonUK
  2. 2.School of Mathematical SciencesUniversity of BathBathUK

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