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Smooth Schur factorizations in the continuation of separatrices. (English) Zbl 1110.65120

The author applies smooth block Schur factorizations of matrix functions to the numerical computation of separatrices in dynamical systems. He implements and improves the numerical algorithms OF L. Dieci and J. Rebaza [BIT 44, 41–62 (2004; Zbl 1052.65113); Erratum ibid. 44, 617–618 (2004; Zbl 1075.65145)].

MSC:

65P30 Numerical bifurcation problems
15A23 Factorization of matrices
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37M20 Computational methods for bifurcation problems in dynamical systems

Software:

AUTO; HomCont
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Full Text: DOI

References:

[1] Dieci, L.; Eirola, T., On smooth decomposition of matrices, SIAM J. Matrix Anal. Appl., 20, 3, 800-819 (1999) · Zbl 0930.15014
[2] Dieci, L.; Friedman, M., Continuation of invariant subspaces, Numer. Linear Algebra Appl., 8, 317-327 (2001) · Zbl 1055.65054
[3] Beyn, W. J., On well-posed problems for connecting orbits in dynamical systems, Contemp. Math., 172, 131-168 (1994) · Zbl 0813.58046
[4] Dieci, L.; Rebaza, J., Point-to-periodic and periodic-to-periodic connections, BIT Numer. Math., 44, 41-62 (2004) · Zbl 1052.65113
[5] Rebaza, J., Computation of traveling waves via an algorithm for connecting orbits, Commun. Appl. Nonlinear Anal., 13, 4, 43-59 (2006) · Zbl 1122.65093
[6] E. Doedel, A. Champneys, T. Fairgrieve, R. Paffenroth, Y. Kuznetsov, B. Sandstede, X.J. Wang, AUTO 2000: Continuation and bifurcation software for ordinary differential equations (with HomCont), Technical Report, 2001. Available from: <http://indy.cs.concordia.ca/auto>; E. Doedel, A. Champneys, T. Fairgrieve, R. Paffenroth, Y. Kuznetsov, B. Sandstede, X.J. Wang, AUTO 2000: Continuation and bifurcation software for ordinary differential equations (with HomCont), Technical Report, 2001. Available from: <http://indy.cs.concordia.ca/auto>
[7] Katzengruber, B.; Krupa, M.; Szmolyan, P., Bifurcation of traveling waves in extrinsic semiconductors, Physica D, 144, 1-19 (2000) · Zbl 0960.34031
[8] Serban, R.; Koon, W. S.; Lo, M.; Marsden, J. E.; Petzold, L. R.; Ross, S. D.; Wilson, R. S., Halo orbit mission correction maneuvers using optimal control, Automatica, 38, 571-583 (2002) · Zbl 1172.93388
[9] Aronson, D.; Doedel, E.; Othmer, H., An analytical and numerical study of the bifurcations in a system of linearly coupled oscillators, Physica, 25D, 20-104 (1987) · Zbl 0624.34029
[10] Wilczak, D., Symmetric heteroclinic connections in the Michelson system – a computer-assisted proof, SIAM J. Appl. Dynam. Systems, 4, 3, 489-514 (2005) · Zbl 1120.34033
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