On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations

Communications in Nonlinear Science and Numerical Simulation

Volumn 16, Issue 3, 2011, Pages 1541-1554

  Butcher, Eric A ^a   Bobrenkov, Oleg A ^a  

^a New Mexico State Univ   (United States)

Author keywords

Chebyshev collocation; Continuous time approximation; Periodic delay differential equations

Indexed keywords

CONTINUOUS TIME SYSTEMS; CONTROL NONLINEARITIES; NONLINEAR EQUATIONS; ORDINARY DIFFERENTIAL EQUATIONS;

APPROXIMATION TECHNIQUES; CHEBYSHEV; CHEBYSHEV COLLOCATION; CONSTANT COEFFICIENTS; CONTINOUS TIME; CONTINUOUS TIME APPROXIMATION; DELAY DIFFERENTIAL EQUATIONS; DELAYED SYSTEMS; PERIODIC DELAY DIFFERENTIAL EQUATION; SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS;

LINEAR SYSTEMS;

EID: 77957355390     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2010.05.037     Document Type: Article

References (25)

1. Breda D., Maset S., Vermiglio R. Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM J Sci Comput 2005, 27:482-495.
2. Trefethen L.N. Spectral methods in Matlab 2000, SIAM Press, Philadelphia.
3. Sun J.-Q. A method of continuous time approximation of delayed dynamical systems. Commun Nonlinear Syst Numer Simul 2009, 14:998-1007. 10.1016/j.cnsns.2008.02.008.
4. Sun J.-Q., Song B. Control studies of time-delayed dynamical systems with the method of continuous time approximation. Commun Nonlinear Syst Numer Simul 2009, 14:3933-3944. 10.1016/j.cnsns.2009.02.011.
5. Breda D., Maset S., Vermiglio R. Computing the characteristic roots for delay differential equations. IMA J Numer Anal 2004, 24:1-19.
6. Fox L., Parker I.B. Chebyshev polynomials in numerical analysis 1968, Oxford University Press, London.
7. Butcher E.A., Bobrenkov O.A., Bueler E., Nindujarla P. Analysis of milling stability by the Chebyshev collocation method: algorithm and optimal stable immersion levels. J Comput Nonlinear Dynam 2009, 4:031003.
8. Bueler E. Error bounds for approximate eigenvalues of periodic-coefficient linear delay differential equations. SIAM J Numer Anal 2007, 45(6):2510-2536.
9. Breda D, Maset S, Vermiglio R. Numerical computation of characteristic multipliers for linear time-periodic delay differential equations. In: Proceedings of the sixth IFAC workshop on linear time-delay systems, L'Aquila, Italy; 2006.
10. Insperger T., Stépán G. Semi-discretization method for delayed systems. Int J Numer Methods Eng 2002, 55(5):503-518.
11. Bayly P.V., Halley J.E., Mann B.P., Davis M.A. Stability of interrupted cutting by temporal finite element analysis. J Manuf Sci Eng 2003, 125:220-225.
12. Bátkai A., Piazzera S. Semigroups for delay equations. Research notes in mathematics 2005, vol. 10. A K Peters Ltd., Wellesley, MA.
13. Michiels W., Niculescu S.-I. Stability and stabilization of time-delay systems: an eigenvalue-based approach 2008, SIAM Press, Philadelphia.
14. Hale J.K., Verduyn Lunel S. Introduction to functional differential equations 1993, Springer-Verlag, New York.
15. Breda D. Solution operator approximations for characteristic roots of delay differential equations. Appl Numer Math 2006, 56:305-317.
16. Butcher EA, Bobrenkov OA. The Chebyshev spectral continuous time approximation for periodic delay differential equations. In: Proceedings of IDETC/CIE ASME conference, San Diego, CA, USA; 2009.
17. Bellen A. One-step collocation for delayed differential equations. J Comp Appl Math 1984, 10:275-283.
18. Ito K., Tran H.T., Manitius A. A fully-discrete spectral method for delay-differential equations. SIAM J Numer Anal 1991, 28:1121-1140.
19. Engelborghs K., Luzyanina T., In 't Hout K.J., Roose D. Collocation methods for the computation of periodic solutions of delay differential equations. SIAM J Sci Comput 2000, 22:1593-1609.
20. Niculescu S.-I. Delay effects on stability: a robust control approach 2001, Springer-Verlag, London.
21. Hsu C.S., Bhatt S.J. Stability charts for second order dynamical systems with time lag. J Appl Mech (ASME) 1966, 33E(1):119-124.
22. Insperger T., Stépán G. Stability chart for the delayed Mathieu equation. Proc R Soc Math Physic Eng Sci 2002, 458:1989-1998.
23. Mathieu É. Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique. J Math 1868, 13:137-203.
24. Van der Pol F., Strutt M.J.O. On the stability of the solutions of Mathieu's equation. Philos Mag J Sci 1928, 5:18-38.
25. Diekmann O., van Gils S.A., Verduyn Lunel S.M., Walther H.O. Delay equations - functional, complex and nonlinear analysis. AMS series 1995, 110. Springer-Verlag, New York.