Floquet theory as a computational tool

SIAM Journal on Numerical Analysis

Volumn 42, Issue 6, 2005, Pages 2522-2568

  Moore, Gerald ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Continuation; Floquet; Fourier; Period doubling; Periodic orbit

Indexed keywords

APPROXIMATION THEORY; BIFURCATION (MATHEMATICS); EIGENVALUES AND EIGENFUNCTIONS; LINEAR ALGEBRA; MATRIX ALGEBRA;

CONTINUATION; FLOQUET; FOURIER; PERIODIC ORBIT;

ALGORITHMS;

EID: 29144517441     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903434175     Document Type: Article

References (30)

1. H. AMANN, Ordinary Differential Equations, Walter de Gruyter, Berlin, 1990.
2. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. Du CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, 2nd ed., SIAM, Philadelphia, 1995.
3. U. M. ASCHER, R. M. MATTHEIJ, AND R. D. RUSSELL, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, 1995.
4. R. H. BARTELS AND G. W. STEWART, Solution of the equation AX + XB = C, Comm. ACM, 15 (1972), pp. 820-826.
5. W. J. BEYN, On well-posed problems for connecting orbits in dynamical systems, in Chaotic Numerics, Contemp. Math. 172, P. E. Kloeden and K. J. Palmer, eds., AMS, Providence, RI, 1994, pp. 131-168.
6. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.
7. F. CHATELIN, Eigenvalues of Matrices, John Wiley, Chichester, UK, 1993.
8. S-N. CHOW AND J. K. HALE, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.
9. M. DELLNITZ, A computational method and path following for periodic solutions with symmetry, in Continuation and Bifurcations: Numerical Techniques and Applications, D. Roose, B. De Dier, and A. Spence, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990, pp. 153-167.
10. M. DELLNITZ, Computational bifurcation of periodic solutions in systems with symmetry, IMA J. Numer. Anal., 12 (1992), pp. 429-455.
11. E. J. DOEDEL, A. D. JEPSON, AND H. B. KELLER, Numerical methods for Hopf bifurcation and continuation of periodic solution paths, in Computing Methods in Applied Sciences and Engineering, Vol. VI, R. Glowinski and J. L. Lions, eds., North-Holland, Amsterdam, 1984, pp. 127-138.
12. E. J. DOEDEL, H. B. KELLER, AND J. P. KERNÉVEZ, Numerical analysis and control of bifurcation problems. Part 1: Bifurcation in finite dimensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1 (1991), pp. 493-520.
13. E. J. DOEDEL, H. B. KELLER, AND J. P. KERNÉVEZ, Numerical analysis and control of bifurcation problems. Part 2: Bifurcation in infinite dimensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1 (1991), pp. 745-772.
14. E. J. DOEDEL, A. R. CHAMPNEYS, T. F. FAIRGRIEVE, YU. A. KUZNETSOV, B. SANDSTEDE, AND X. J. WANG, Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), Technical report, California Institute of Technology, Pasadena, CA, 1998; also available online from http://indy.cs.concordia.ca/auto/main.html.
15. E. J. DOEDEL, W. GOVAERTS, AND YU. A. KUZNETSOV, Computation of periodic solution bifurcations in ODEs using bordered systems, SIAM J. Numer. Anal., 41 (2003), pp. 401-435.
16. E. W. GEKELER, On trigonometric collocation in Hopf bifurcation, in Bifurcation and Symmetry, E. Allgower, K. Böhmer, and M. Golubitsky, eds., Internat. Ser. Numer. Math. 104, Birkhäuser, Basel, Switzerland, 1992, pp. 147-156.
17. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
18. W. J. GOVAERTS, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000.
19. R. GRIMSHAW, Nonlinear Ordinary Differential Equations, Blackwell, Oxford, UK, 1990.
20. J. HALE, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.
21. G. IOOSS AND D. D. JOSEPH, Elementary Stability and Bifurcation Theory, 2nd ed., Springer-Verlag, New York, 1990.
22. T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
23. H. B. KELLER AND A. D. JEPSON, Steady state and periodic solution paths: Their bifurcations and computations, in Numerical Methods for Bifurcation Problems, T. Küpper, H. D. Mittelmann, and H. Weber, eds., Internat. Schriftenreihe Numer. Math. 70, Birkhäuser, Basel, Switzerland, 1984, pp. 219-246.
24. G. MOORE, Computation and parametrization of periodic and connecting orbits, IMA J. Numer. Anal., 15 (1995), pp. 245-263.
25. G. MOORE, Computation and parametrisation of invariant curves and tori, SIAM J. Numer. Anal., 33 (1996), pp. 2333-2358.
26. G. MOORE, Laguerre approximation of stable manifolds with application to connecting orbits, Math. Comp., 73 (2004), pp. 211-242.
27. R. SEYDEL, From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis, Elsevier, New York, 1988.
28. G. W. STEWART, Error and perturbation bounds for subspaces associated with certain eigen-value problems, SIAM Rev., 15 (1973), pp. 727-764.
29. C. F. VAN LOAN, Computational Frameworks for the Fast Fourier Transform, SIAM, Philadelphia, 1992.
30. D. VISWANATH, The Lindstedt-Poincaré technique as an algorithm for computing periodic orbits, SIAM Rev., 43 (2001), pp. 478-495.