Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation

Nonlinear Dynamics

Volumn 52, Issue 1-2, 2008, Pages 137-149

  Deshmukh, Venkatesh ^a   Butcher, Eric A ^b   Bueler, Ed ^c  

^a Villanova University   (United States)

^b Horeshoe Cir Jett Hall   (United States)

^c UNIVERSITY OF ALASKA FAIRBANKS   (United States)

Author keywords

Center manifold reduction; Chebyshev spectral collocation; Delay differential equations; Periodic systems

Indexed keywords

CHEBYSHEV APPROXIMATION; COMPUTER SIMULATION; NONLINEAR EQUATIONS; PERTURBATION TECHNIQUES; TIME DELAY;

CHEBYSHEV SPECTRAL COLLOCATION; DELAY DIFFERENTIAL EQUATIONS; DIMENSIONAL REDUCTION; PERIODIC COEFFICIENTS; PERTURBATION EXPANSION;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 39749182700     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-007-9266-6     Document Type: Article

References (33)

1. Hassard, B., Wan, Y.H.: Bifurcation formulae derived from center manifold. J. Math. Anal. Appl. 63, 297-312 (1978)
2. Hassard, B.D., Kazarinoff, N.D., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Notes Series, vol. 41. Cambridge University Press, Cambridge (1981)
3. Carr, J.: Applications of Centre Manifold Theory. Springer, New York (1981)
4. Kalmar-Nagy, T., Stepan, G., Moon, F.C.: Subcritical Hopf bifucration in the delay equation model for machine tool vibrations. Nonlinear Dyn. 26, 121-142 (2001)
5. Gilsinn, D.E.: Estimating critical Hopf bifurcation parameters for a second order delay differential equation with application to machine tool chatter. Nonlinear Dyn. 30, 103-154 (2002)
6. Wang, Z.H., Hu, H.Y.: Dimensional reduction for nonlinear time-delayed systems composed of stiff and soft substructures. Nonlinear Dyn. 25, 317-331 (2001)
7. Das, S.L., Chaterjee, A.: Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dyn. 30, 323-335 (2002)
8. Wahi, P., Chaterjee, A.: Galerkin projections for delay differential equations. In: Proceedings of ASME DETC'03/VIB-48570, Chicago, 2-6 September 2003
9. Shampine, L.F., Thompson, F.: Solving DDEs in MATLAB. Appl. Numer. Math. 37, 441-458 (2001)
10. Engelborghs, K., Luzyanina, T., Samaey, G.: DDE-BIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations, TW 330. Department of Computer Science, Katholieke Universiteit Leuven, Belgium (2001)
11. Maccari, A.L.: Response of a parametrically excited van der Pol oscillator to a time delay state feedback. Nonlinear Dyn. 26, 105-119 (2001)
12. Ma, H., Butcher, E.A., Bueler, E.: Chebyshev expansion of linear and piecewise linear dynamic systems with time delay and periodic coefficients under control excitations. ASME J. Dyn. Syst. Meas. Control 125, 236-243 (2003)
13. Faria, T.: Normal forms for periodic retarded functional equations. Proc. Roy. Soc. Edinb. A 127, 21-46 (1997)
14. Szalai, R., Stepan, G., Hogan, S.J.: Continuation of bifurcations in periodic DDEs using characteristic matrices. SIAM J. Appl. Math. (in press), available at http://www.enm.bris.ac.uk/anm/preprints/2004r23.html
15. Szalai, R., Stepan, G., Hogan, S.J.: Global dynamics of low-immersion high-speed milling. Chaos 14, 1-9 (2004)
16. Stepan, G., Szalai, R., Mann, B., Bayly, P., Insperger, T., Gradisek, J., Govekar, E.: Nonlinear dynamics of high-speed milling-analyses, numerics, and experiments. J. Vib. Acoust. 127, 197-203 (2005)
17. Szalai, R., Stepan, G.: Period doubling bifurcation and center manifold reduction in a time-periodic and time-delayed model of machining. J. Nonlinear Sci. (in press)
18. Bessman, J.W.: The "adjoint" in linear functional differential equations. PhD dissertation, Catholic University of America (1968)
19. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)
20. Chicone, C., Latushkin, Y.: Center manifold for infinite dimensional nonautonomous differential equations. J. Differ. Equ. 141, 356-399 (1997)
21. Rost, G.: Neimark-Sacker bifurcation for periodic delay differential equations. Nonlinear Anal. 60, 1025-1044 (2005)
22. Halanay, S.: Differential Equations: Stability, Oscillations, Time Lags. Academic, New York (1966)
23. Butcher, E.A., Ma, H., Bueler, E., Averina, V., Szabo, Z.: Stability of linear time-periodic delay-differential equations via Chebyshev polynomials. Int. J. Numer. Methods Eng. 59, 895-922 (2004)
24. Bayly, P.V., Halley, J.E., Davies, M.A., Mann, B.P.: Stability of interrupted cutting by time finite element analysis. J. Manuf. Sci. Eng. 125, 220-225 (2003)
25. Gilsinn, D.E., Potra, F.A.: Integral operators and delay differential equations. J. Integral Equ. Appl. 18(3), 297-336 (2006)
26. 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. 22, 1593-1609 (2000)
27. Bueler, E.: A Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates. arXiv:math.NA/0409464 (2004)
28. Butcher, E.A., Nindujarla, P., Bueler, E.: Stability of up- and down-milling using Chebyshev collocation method. In: Proceedings ASME DETC 2005-84880, Long Beach, 24-28 September 2005
29. Trefethen, L.N.: Spectral Methods in MATLAB, Software, Environments, and Tools. SIAM, Philadelphia (2000)
30. Wiggins, S.: Introduction to Applied Nonlinear Systems and Chaos. Springer, New York (2003)
31. Butcher, E.A., Sinha, S.C.: Symbolic computation of local stability and bifurcation surfaces for nonlinear time-periodic systems. Nonlinear Dyn. 17, 1-21 (1998)
32. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
33. David, A., Sinha, S.C.: Versal deformation and local bifurcation analysis of time-periodic nonlinear systems. Nonlinear Dyn. 21, 317-336 (2000)