×

zbMATH — the first resource for mathematics

Periodic solutions of nonlinear delay differential equations using spectral element method. (English) Zbl 1246.65102
Summary: We extend the temporal spectral element method further to study the periodic orbits of general autonomous nonlinear delay differential equations (DDEs) with one constant delay. Although we describe the approach for one delay to keep the presentation clear, the extension to multiple delays is straightforward. We also show the underlying similarities between this method and the method of collocation. The spectral element method that we present here can be used to find both the periodic orbit and its stability. This is demonstrated with a variety of different examples, namely, the delayed versions of Mackey-Glass equation, Van der Pol equation, and Duffing equation. For each example, we show the method’s convergence behavior using both \(p\) and \(h\) refinement and we provide comparisons between equal size meshes that have different distributions.

MSC:
65L03 Numerical methods for functional-differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
Software:
AUTO; RADAR5
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atay, F.: Van der Pol’s oscillator under delayed feedback. J. Sound Vib. 218(2), 333–339 (1998) · Zbl 1235.70142 · doi:10.1006/jsvi.1998.1843
[2] Baker, C., Bocharov, G., Ford, J., Lumb, P., Norton, S., Paul, C., Junt, T., Krebs, P., Ludewig, B.: Computational approaches to parameter estimation and model selection in immunology. J. Comput. Appl. Math. 184(1), 50–76 (2005). doi: 10.1016/j.cam.2005.02.003 . Special Issue on Mathematics Applied to Immunology · Zbl 1072.92020 · doi:10.1016/j.cam.2005.02.003
[3] Barton, D., Krauskopf, B., Wilson, R.: Collocation schemes for periodic solutions of neutral delay differential equations. J. Differ. Equ. Appl. 12(11), 1087–1101 (2006). doi: 10.1080/10236190601045663 · Zbl 1108.65089 · doi:10.1080/10236190601045663
[4] Barton, D., Krauskopf, B., Wilson, R.: Homoclinic bifurcations in a neutral delay model of a transmission line oscillator. Nonlinearity 20(4), 809–829 (2007). doi: 10.1088/0951-7715/20/4/001 · Zbl 1129.34049 · doi:10.1088/0951-7715/20/4/001
[5] Bellen, A., Zennaro, M.: Numerical Solution of Delay Differential Equations. Oxford University Press, Oxford (2003) · Zbl 1038.65058
[6] Berrut, J., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004) · Zbl 1061.65006 · doi:10.1137/S0036144502417715
[7] Bobrenkov, O., Khasawneh, F., Butcher, E., Mann, B.: Analysis of milling dynamics for simultaneously engaged cutting teeth. J. Sound Vib. 329(5), 585–606 (2010). doi: 10.1016/j.jsv.2009.09.032 · doi:10.1016/j.jsv.2009.09.032
[8] Bocharov, G.A., Rihan, F.A.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125(1-2), 183–199 (2000). doi: 10.1016/S0377-0427(00)00468-4 · Zbl 0969.65124 · doi:10.1016/S0377-0427(00)00468-4
[9] Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2001) · Zbl 0994.65128
[10] Butcher, E., Bobrenkov, O., Bueler, E., Nindujarla, P.: Analysis of milling stability by the Chebyshev collocation method: Algorithm and optimal stable immersion levels. J. Comput. Nonlinear Dyn. 4(3), 031003 (2009). doi: 10.1115/1.3124088
[11] Doedel, E., Champneys, A., Fairgrieve, T., Kuznetsov, Y., Sandstede, B., Wang, X.: Auto 97: Continuation and bifurcation software for ordinary differential equations. Available online at: http://indy.cs.concordia.ca/auto/ (1998)
[12] Doedel, E.J., Keller, H.B., Kernévez, J.P.: Numerical analysis and control of bifurcation problems: II. Bifurcation in infinite dimensions. Int. J. Bifurc. Chaos 1(4), 745–772 (1991). doi: 10.1142/S0218127491000555 · Zbl 0876.65060 · doi:10.1142/S0218127491000555
[13] Engelborghs, K., Doedel, E.: Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations. Numer. Math. 91, 627–648 (2002) · Zbl 1002.65089 · doi:10.1007/s002110100313
[14] Engelborghs, K., Luzyanina, T., ’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) · Zbl 0981.65082 · doi:10.1137/S1064827599363381
[15] Eslahchi, M., Masjed-Jamei, M., Babolian, E.: On numerical improvement of Gauss-Lobatto quadrature rules. Appl. Math. Comput. 164(3), 707–717 (2005). doi: 10.1016/j.amc.2004.04.113 · Zbl 1070.65019 · doi:10.1016/j.amc.2004.04.113
[16] Guglielmi, N., Hairer, E.: Implementing Radau IIA methods for stiff delay differential equations. Computing 67(1), 1–12 (2001) · Zbl 0986.65069 · doi:10.1007/s006070170013
[17] Guglielmi, N., Hairer, E.: Users’ guide for the code RADAR5–version 2.1. Tech. rep., Università dell’Aquila, Italy (2005)
[18] Guglielmi, N., Hairer, E.: Computing breaking points in implicit delay differential equations. Adv. Comput. Math. 29(3), 229–247 (2008) · Zbl 1160.65041 · doi:10.1007/s10444-007-9044-5
[19] Hale, J., Sternberg, N.: Onset of chaos in differential delay equations. J. Comput. Phys. 77(1), 221–239 (1988). doi: 10.1016/0021-9991(88)90164-7 · Zbl 0644.65050 · doi:10.1016/0021-9991(88)90164-7
[20] Hale, J.K., Lunel, S.V.: Introduction to Functional Differential Equations. Springer, New York (1993) · Zbl 0787.34002
[21] an der Heiden, U.: Unfolding complexity: hereditory dynamical systems–new bifurcation schemes and high dimensional chaos. In: Nonlinear Dynamics and Chaos: Where Do We Go From Here? IoP, Bristol (2003)
[22] Higham, N.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24(4), 547–556 (2004). doi: 10.1093/imanum/24.4.547 · Zbl 1067.65016 · doi:10.1093/imanum/24.4.547
[23] Hu, H., Dowell, E.H., Virgin, L.N.: Resonances of a harmonically forced duffing oscillator with time delay state feedback. Nonlinear Dyn. 15(4), 311–327 (1998). doi: 10.1023/A:1008278526811 · Zbl 0906.34052 · doi:10.1023/A:1008278526811
[24] Khasawneh, F., Mann, B.: A spectral element approach for the stability of delay systems. Int. J. Numer. Methods Eng. (2011). doi: 10.1002/nme.3122 · Zbl 1242.70007
[25] Khasawneh, F., Mann, B., Insperger, T., Stépán, G.: Increased stability of low-speed turning through a distributed force and continuous delay model. J. Comput. Nonlinear Dyn. 4(4), 041003 (2009)
[26] Krauskopf, B.: Bifurcation analysis of lasers with delay. In: Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, pp. 147–183, Wiley, New Jersey (2005)
[27] Lin, G.: Periodic solutions for Van der Pol equation with time delay. Appl. Math. Comput. 187(2), 1187–1198 (2007). doi: 10.1016/j.amc.2006.09.032 · Zbl 1210.34096 · doi:10.1016/j.amc.2006.09.032
[28] Luzyanina, T., Engelborghs, K.: Computing Floquet multipliers for functional differential equations. Int. J. Bifurc. Chaos 12(12), 2977–2989 (2002) · Zbl 1048.65126 · doi:10.1142/S0218127402006291
[29] Mackey, M., Glass, L.: Oscillation and chaos in physiological control systems. Science 197(4300), 287–289 (1977). doi: 10.1126/science.267326 · Zbl 1383.92036 · doi:10.1126/science.267326
[30] Mann, B.P., Sims, N.D.: Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound Vib. 319(1-2), 515–530 (2009) · doi:10.1016/j.jsv.2008.06.011
[31] Mann, B.P., Young, K.A.: An empirical approach for delayed oscillator stability and parametric identification. Proc. R. Soc. A 462, 2145–2160 (2006) · Zbl 1149.70323 · doi:10.1098/rspa.2006.1677
[32] Parter, S.: On the Legendre–Gauss–Lobatto points and weights. J. Sci. Comput. 14(9), 347–355 (1999) · Zbl 0959.65113 · doi:10.1023/A:1023204631825
[33] Patera, A.: A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984) · Zbl 0535.76035 · doi:10.1016/0021-9991(84)90128-1
[34] Radde, N.: The impact of time delays on the robustness of biological oscillators and the effect of bifurcations on the inverse problem. EURASIP Journal on Bioinformatics and Systems. Biology 2009, 1–14 (2009). doi: 10.1155/2009/327503
[35] Reddy, J.: An Introduction to the Finite Element Method, 2nd edn. McGraw-Hill, New York (1993)
[36] Seydel, R.: Practical Bifurcation and Stability Analysis, 3rd edn. Springer, Berlin (2009). doi: 10.1007/978-1-4419-1740-9 · Zbl 1195.34004
[37] Shahverdiev, E., Bayramov, P., Shore, K.: Cascaded and adaptive chaos synchronization in multiple time-delay laser systems. Chaos Solitons Fractals 42(1), 180–186 (2009). doi: 10.1016/j.chaos.2008.11.004 · doi:10.1016/j.chaos.2008.11.004
[38] Sims, N.D., Mann, B., Huyanan, S.: Analytical prediction of chatter stability for variable pitch and variable helix milling tools. J. Sound Vib. 317, 664–686 (2008) · doi:10.1016/j.jsv.2008.03.045
[39] Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. Wiley, New York (1989) · Zbl 0686.34044
[40] Szalai, R., Stépán, G.: Lobes and lenses in the stability chart of interrupted turning. J. Comput. Nonlinear Dyn. 1, 205–211 (2006) · doi:10.1115/1.2198216
[41] Vu, T.H., Deeks, A.J.: Use of higher-order shape functions in the scaled boundary finite element method. Int. J. Numer. Methods Eng. 65, 1714–1733 (2006) · Zbl 1115.74053 · doi:10.1002/nme.1517
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.