×

zbMATH — the first resource for mathematics

Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations. (English) Zbl 0910.34057
A new numerical method is presented for efficient computation of periodic solutions to nonlinear systems of delay differential equations (DDEs) with several discrete delays. This method exploits typical spectral properties of the monodromy matrix of a DDE – most Floquet multipliers are very close to zero and allow an effective computation of the dominant Floquet multipliers to determine the stability of a periodic solution. It basically combines Newton-based single shooting for weakly stable and unstable modes with time integration or a simple Picard scheme in the orthogonal complement. The method is applied to two examples and is shown to give precise and reliable results.
Reviewer: A.Steindl (Wien)

MSC:
34K10 Boundary value problems for functional-differential equations
65Q05 Numerical methods for functional equations (MSC2000)
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
34C25 Periodic solutions to ordinary differential equations
Software:
PDDE-CONT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1137/0316013 · Zbl 0379.49025 · doi:10.1137/0316013
[2] DOI: 10.1137/0147039 · Zbl 0627.34046 · doi:10.1137/0147039
[3] DOI: 10.1016/0022-0396(74)90084-9 · Zbl 0295.34055 · doi:10.1016/0022-0396(74)90084-9
[4] Chow S.-N., Trans. Amer. Math. Soc. 370 (1) pp 127– (1988)
[5] Doedel E. J., Cong. (34) pp 225– (1982)
[6] DOI: 10.1016/0096-3003(91)90093-3 · Zbl 0729.65052 · doi:10.1016/0096-3003(91)90093-3
[7] DOI: 10.1007/BF01403681 · Zbl 0419.34070 · doi:10.1007/BF01403681
[8] DOI: 10.1016/0022-247X(74)90162-0 · Zbl 0293.34102 · doi:10.1016/0022-247X(74)90162-0
[9] DOI: 10.1137/0506028 · Zbl 0241.34080 · doi:10.1137/0506028
[10] Liu X., Dynam. Syst. Appl. 3 pp 357– (1994)
[11] Lust K., SIAM J. Sci. Comput. (1997)
[12] DOI: 10.1016/0377-0427(96)00008-8 · Zbl 0855.65092 · doi:10.1016/0377-0427(96)00008-8
[13] DOI: 10.1142/S0218127496000333 · Zbl 0875.93183 · doi:10.1142/S0218127496000333
[14] Nussbaum R. D., I. Ann. Mat. Pura Appl. 101 pp 236– (1974)
[15] DOI: 10.1137/0146013 · Zbl 0604.35039 · doi:10.1137/0146013
[16] DOI: 10.1016/0960-0779(95)90873-Q · Zbl 1080.65542 · doi:10.1016/0960-0779(95)90873-Q
[17] DOI: 10.1137/0730057 · Zbl 0789.65037 · doi:10.1137/0730057
[18] Walther H.-O., J. Diff. Eqns. 39 pp 369– (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.