Continuation of invariant subspaces in large bifurcation problems

SIAM Journal on Scientific Computing

Volumn 30, Issue 2, 2007, Pages 637-656

  Bindel, David ^a   Demmel, James ^b   Friedman, Mark ^c  

^a NEW YORK UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF ALABAMA IN HUNTSVILLE   (United States)

Author keywords

Bifurcation analysis; Continuation; Invariant subspaces; Ritz galerkin methods; Smooth eigendecompositions

Indexed keywords

BIFURCATION (MATHEMATICS); GALERKIN METHODS;

BIFURCATION ANALYSIS; BIFURCATION PROBLEM; CONTINUATION; CONTINUATION OF INVARIANT SUBSPACES; EIGEN DECOMPOSITION; INVARIANT SUBSPACE; ORTHONORMAL BASIS; PARAMETER-DEPENDENT MATRIX; RITZ-GALERKIN METHODS; SMOOTH EIGENDECOMPOSITION;

SOFTWARE TESTING;

EID: 55549112582     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060654219     Document Type: Article

References (44)

1. P.-A. ABSIL, R. SEPULCHRE, P. VAN DOOREN, AND R. MAHONY, Cubically convergent iterations for invariant subspace computation, SIAM J. Matrix Anal. Appl, 26 (2004), pp. 70-96.
2. E. ANDERSON, Z. BAI, C. BISCHOF, S. BLACKFORD, J. DEMMEL, J. DONGARRA, J. D. CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, ANDD. SORENSEN, LAPACK Users' Guide, 3rd ed., SIAM, Philadelphia, 1999.
3. P. ASHWIN, K. BOHMER, AND Z. MEI, A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square, Math. Comp., 64 (1995), pp. 649-670, S19-S22.
4. P. ASHWIN AND Z. MEI, A Hopf bifurcation with Robin boundary conditions, J. Dynam. Differential Equations, 6 (1994), pp. 487-503.
5. W.-J. BEYN, A. CHAMPNEYS, E. J. DOEDEL, Y. A. KUZNETSOV, B. SANDSTEDE, AND W. GO-VAERTS, Numerical continuation and computation of normal forms, in Handbook of Dynamical Systems III: Towards Applications, B. Fiedler, ed., Elsevier, Amsterdam, 2001, pp. 140-219.
6. W.-J. BEY'N, W. KLESS, AND V. THÜMMLER, Continuation of low-dimensional invariant subspaces in dynamical systems of large dimension, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler, ed., Springer, Berlin, 2001, pp. 47-72.
7. D. BINDEL, J. DEMMEL, AND M. FRIEDMAN, Continuation of invariant subspaces for large bifurcation problems, in Proceedings of the SIAM Conference on Linear Algebra, Williamsburg, VA, 2003.
8. D. BINDEL, W. DEMMEL, M. FRIEDMAN, W. GOVAERTS, AND Y. A. KUZNETSOV, Bifurcation analysis of large equilibrium systems in MATLAB, in Proceedings of the ICCS conference 2005, Atlanta, GA, 2005, pp. 50-57.
9. J. BOSEC, Continuation of Invariant Subspaces in Bifurcation Problems, Ph.D. thesis, University of Marburg, Marburg, Germany, 2002.
10. J. H. BRANDTS, The Riccati algorithm for eigenvalues and invariant subspaces of matrices with inexpensive action, Linear Algebra Appl. 358 (2003), pp. 335-365.
11. E. A. BURROUGHS, R. B. LEHOUCQ, L. A. ROMERO, ANDA. J. SALINGER, Linear Stability of Flow in a Differentially Heated Cavity via Large-Scale Eigenvalue Calculations, Technical report SAND2002-3036J, Sandia National Laboratories, Livermore, CA, 2002.
12. C. S. CHIEN AND M. H. CHEN, Multiple bifurcations in a reaction-diffusion problem, Comput. Math. Appl., 35 (1998), pp. 15-39.
13. C. S. CHIEN, Z. MEI, AND C. L. SHEN, Numerical continuation at, double bifurcation points of a reaction-diffusion problem, Internat J. Bifur. Chaos Appl. Sci. Engrg., 8 (1997), pp. 117-139.
14. K. A. CLIFFE, A. SPENCE, AND S. J. TAVENER, The numerical analysis of bifurcations problems vith application to fluid mechanics, Acta Numer., 9 (2000), pp. 39-131.
15. G. DANGELMAYR, Degenerate bifurcations near a double eigenvalue in the Brusselator, J, Austral. Math. Soc. Ser. B, 28 (1987), pp. 486-535.
16. J. W. DEMMEL, Three methods for refining estimates of invariant subspaces, Computing, 38 (1987), pp. 43-57.
17. J. W. DEMMEL, L. DIECI, AND M. J. FREIDMAN, Computing connecting orbits via an improved algorithm for continuing invariant subspaces, SIAM J. Sci. Comput., 22 (2001), pp. 81-94.
18. A. DHOOGE, W. GOVAERTS, Y. KUZNETSOV, W. MESTROM, AND A. M. RIET, MATLAB continuation software package CL-MATGONT, Jan. 2003, available online at http://www.math.uu.nl/people/kuznet/cm/.
19. L. DLECI AND T. EIROLA, On smooth decompositions of matrices, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 800-819.
20. L. DIECI AND M. J. FRIEDMAN, Continuation of invariant subspaces, Numer. Linear Algebra. Appl., 8 (2001), pp. 317-327.
21. L. DIECI AND A. PAPINI, Continuation of eigendecompositions. Future Generation Computer Systems, 19 (2003), pp. 1125-1137.
22. L. DIECI AND A. J. REBAZA, Point-to-periodic and periodic-to-periodic connections, BIT, 44 (2004), pp. 41-60.
23. E. J. DOEDEL AND H. SHARIFI, Collocation methods for continuation problems in nonlinear elliptic PDEs, Notes Numer. Fluid Mech. 74 (2000), pp. 105-118.
24. A. EDELMAN, T. ARIAS, AND S. SMITH, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 303-353.
25. M. FRIEDMAN, W. GOVAERTS, Y. KUZNETSOV, AND B. SAUTOIS, Continuation of homoclinic orbits in MATLAB, in Proceedings of the ICCS Conference 2005, Atlanta, GA, 2005, pp. 263-270.
26. M. J. FRIEDMAN, Improved detection of bifurcations in large nonlinear systems via the continuation of invariant subspaces algorithm, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 2277-2285.
27. M. J. FRIEDMAN AND M. E. JACKSON, An Improved RLV Stability Analysis via a Continuation Approach, Technical, Report, NASA Marshall Space Flight Center, Huntsville, AL, 2002.
28. K. GEORG, Matrix-free numerical continuation and bifurcation, Numer. Funct. Anal. Optim., 22 (2001), pp. 303-320.
29. G. H. GOLUB AND C. F. V. LOAN, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 1989.
30. M. GOLUBITSKY AND D. G. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Vol. 1, Springer-Verlag, New York, 1985.
31. W. GOVAERTS, Computation of singularities in large nonlinear systems, SIAM J. Numer. Anal., 34 (1997), pp. 867-880.
32. W. GOVAERTS, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000.
33. W. GOVAERTS, J. GUCKENHEIMER, AND A. KHIBNIK, Defining functions for multiple Hopf bifurcations, SIAM J. Numer. Anal, 34 (1997), pp. 1269-1288.
34. N. J. HIGHAM, Matrix nearness problems and applications, in Applications of Matrix Theory, M. J. C. Cover and S. Barnett, eds., Oxford University Press, New York, 1989, pp. 1-27.
35. T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, Corrected printing of the second edition, Springer-Verlag, Berlin, 1995.
36. Y. A. KUZNETSOV, Elements of Applied Bifurcation Theory, 2nd ed., Springer-Verlag, New York, 1998.
37. R. LEFEVER. AND I. PRIGOGINE, Symmetry-breaking instabilities in dissipative systems II, J. Chem. Phys., 48 (1968), pp. 1695-1700.
38. Z. MEI, Numerical Bifurcation Analysis for Reaction-Diffusion Equations, Ph.D. thesis, University of Marburg, Marburg, Germany, 1997.
39. D. G. SCHAEFFER AND M. A. GOLUBITSKY, Bifurcation analysis near a double eigenvalue of a model chemical reaction, Arch. Rational Mech. Anal., 75 (1981), pp. 315-347.
40. G. M. SHROFF AND H. B. KELLER, Stabilization of unstable procedures: The recursive projection method, SIAM J. Numer. Anal, 30 (1993), pp. 1099-1120.
41. G. W. STEWART, Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Rev., 15 (1973), pp. 727-764.
42. G. W. STEWART AND J.-G. SUN, Matrix Perturbation Theory, Academic Press, San Diego, CA, 1990.
43. L. N. TREFETHEN, Pseudospectra of matrices, in Numerical Analysis 1991, D. F. Griffiths and G. A. Watson, eds., Longman Sci. Tech., Harlow, Essex, UK, 1991, pp. 234-262.
44. J. M. VARAH, On the separation of two matrices, SIAM J. Numer. Anal., 16 (1979), pp. 212-222.