Bifurcation Analysis of Large Equilibrium Systems in Matlab

  • David S. Bindel
  • James W. Demmel
  • Mark J. Friedman
  • Willy J. F. Govaerts
  • Yuri A. Kuznetsov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3514)


The Continuation of Invariant Subspaces (CIS) algorithm produces a smoothly varying basis for an invariant subspace \(\mathcal{R}(s)\) of a parameter-dependent matrix A(s). In the case when A(s) is the Jacobian matrix for a system that comes from a spatial discretization of a partial differential equation, it will typically be large and sparse. Cl_matcont is a user-friendly matlab package for the study of dynamical systems and their bifurcations. We incorporate the CIS algorithm into cl_atcont to extend its functionality to large scale bifurcation computations via subspace reduction.


  1. 1.
    Allgower, E., Georg, K.: Numerical path following. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 5, pp. 3–207. North-Holland, Amsterdam (1997)Google Scholar
  2. 2.
    Allgower, E., Schwetlick, H.: A general view of minimally extended systems for simple bifurcation points. Z. angew. Math. Mech. 77, 83–98 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bindel, D., Demmel, J., Friedman, M.: Continuation of invariant subspaces for large bifurcation problems. In: Proceedings of the SIAM Conference on Linear Algebra, Williamsburg, VA (2003)Google Scholar
  4. 4.
    Burroughs, E.A., Lehoucq, R.B., Romero, L.A., Salinger, A.J.: Linear stability of flow in a differentially heated cavity via large-scale eigenvalue calculations. Tech. Report SAND2002-3036J, Sandia National Laboratories (2002)Google Scholar
  5. 5.
    Chien, C.S., Chen, M.H.: Multiple bifurcations in a reaction-diffusion problem. Computers Math. Applic. 35, 15–39 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cliffe, K.A., Spence, A., Tavener, S.J.: The numerical analysis of bifurcations problems vith application to fluid mechanics. Acta Numerica, 1–93 (2000)Google Scholar
  7. 7.
    Demmel, J.W., Dieci, L., Friedman, M.J.: Computing connecting orbits via an improved algorithm for continuing invariant subspaces. SIAM J. Sci. Comput. 22, 81–94 (2001)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: matcont: A matlab package for numerical bifurcation analysis of odes. ACM TOMS 29, 141–164 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dhooge, A., Govaerts, W., Kuznetsov, Y.A., Mestrom, W., Riet, A.M.: MATLAB continuation software package CL_MATCONT (January 2003),
  10. 10.
    Dieci, L., Friedman, M.J.: Continuation of invariant subspaces. Numerical Linear Algebra and Appl. 8, 317–327 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Doedel, E.J., Sharifi, H.: Collocation methods for continuation problems in nonlinear elliptic PDEs, issue on continuation. Methods in Fluid Mechanics, Notes on Numer. Fluid. Mech. 74, 105–118 (2000)MathSciNetGoogle Scholar
  12. 12.
    Beyn, W.-J., Kleß, W., Thümmler, V.: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. In: Continuation of low-dimensional Invariant Subspaces in dynamical systems of large dimension, pp. 47–72. Springer, Heidelberg (2001)Google Scholar
  13. 13.
    Govaerts, W.: Numerical methods for bifurcations of dynamical equilibria. SIAM, Philadelphia (2000)MATHCrossRefGoogle Scholar
  14. 14.
    Griewank, A., Reddien, G.: Characterization and computation of generalized turning points. SIAM J. Numer. Anal. 21, 176–185 (1984)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kuznetsov, Y. A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004)MATHGoogle Scholar
  16. 16.
    Schaeffer, D., Golubitsky, M.: Bifurcation analysis near a double eigenvalue of a model chemical reaction. Arch. Rational Mech. Anal. 75, 315–347 (1981)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Shroff, G.M., Keller, H.B.: Stabilization of unstable procedures: The recursive projection method. SIAM J. Numer. Anal. 30, 1099–1120 (1993)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zienkiewicz, O., Taylor, R.T.: The Finite Element Method. Solid Mechanics, 5th edn., vol. 2. Butterworth Heinemann, Oxford (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • David S. Bindel
    • 1
  • James W. Demmel
    • 2
  • Mark J. Friedman
    • 3
  • Willy J. F. Govaerts
    • 4
  • Yuri A. Kuznetsov
    • 5
  1. 1.Department of Electrical Engineering and Computer ScienceUniversity of California at BerkeleyBerkeley
  2. 2.Department of Electrical Engineering and Computer Science, Department of MathematicsUniversity of California at BerkeleyBerkeley
  3. 3.Mathematical Sciences DepartmentUniversity of Alabama in HuntsvilleHuntsville
  4. 4.Department of Applied Mathematics and Computer ScienceGhent UniversityGhentBelgium
  5. 5.Mathematisch InstituutUtrechtThe Netherlands

Personalised recommendations