This paper demonstrates the utility of the high-dimensional harmonic balance (HDHB) method for locating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHB method for systems of DDEs is described in detail. The method has been successfully applied to capture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, the sunflower equation and a circadian rhythm model. The results show excellent agreement with collocation and continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical harmonic balance method are highlighted and discussed.
Asl, F.M. and Ulsoy, A.G., 2003, "Analysis of a system of linear delay differential equations ," Journal of Dynamic Systems, Measurement, and Control125, 215-223.
2.
Beuter, A., Bélair, J., and Labrie, C., 1993, "Feedback and delays in neurological disease: a modeling study using dynamical systems," Bulletin of Mathematical Biology55, 525-541.
3.
Breda, D., Maset, S., and Vermiglio, R., 2004, "Computing the characteristic roots for delay differential equations," IMA Journal of Numerical Analysis24, 1-19.
4.
Butcher, E.A., Ma, H., Bueler, E., Averina, V., and Szabó, Zs. , 2004, "Stability of time-periodic delay-differential equations via Chebyshev polynomials," International Journal for Numerical Methods in Engineering59, 895-922.
5.
Casal, A. and Freedman, M., 1980, "A Poincaré-Lindstedt approach to bifurcation problems for differential delay equations," IEEE Transactions on Automatic Control25(5), 967-973.
6.
Casal, A. and Somolinos, A., 1982, "Forced oscillations for the sunflower equation, entrainment ," Nonlinear Analysis Theory, Methods and Applications6(4), 397-414.
7.
Dowell, E.H. and Hall, K.C., 2001, "Modeling of fluid-structure interaction," Annual Review of Fluid Mechanics33, 413-455.
8.
Engelborghs, K., Luzyanina, T., and Roose, D., 2002, "Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL," ACM Transactions on Mathematical Software28(1), 1-21.
9.
Engelborghs, K., Luzyanina, T., and Samaey, G., 2001, "DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations," Technical Report TW-330, Department of Computer Science, K.U. Leuven, Leuven , Belgium.
10.
Erneux, T. and Kalmár-Nagy, T., 2007, "Nonlinear stability of a delayed feedback controlled container crane," Journal of Vibration and Control13(5), 603.
11.
Gilsinn, D.E., 2005, "Discrete Fourier series approximation to periodic solutions of autonomous delay differential equations," in Proceedings of IDERC/CIE 2005: ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference , Long Beach, CA, September 24-28.
12.
Hall, K.C., Thomas, J.P., and Clark, W.S., 2000, "Computation of unsteady nonlinear flows in cascades using a harmonic balance technique," in 9th International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Lyon, France, September.
13.
Hassard, B.D., Kazarinoff, N.D., and Wan Y.-H., 1981, Theory and Applications of Hopf Bifurcation (London Mathematical Society Lecture Note Series, Vol. 41), Cambridge University Press, Cambridge.
14.
Hu, H.Y., Dowell, E.H., and Virgin, L.N., 1998, "Resonance of a harmonically forced Duffing oscillator with time delay state feedback," Nonlinear Dynamics15(4), 311-327.
15.
Hu, H.Y. and Wang, Z.H., 2002, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer, Berlin.
16.
Insperger T. and Stépán, G., 2002, "Semidiscretization method for general delayed systems ," International Journal for Numerical Methods in Engineering55, 503-518.
17.
Insperger, T., Wahi, P., Colombo, A., Stépán, G., di Bernardo, M., and Hogan, S.J., 2010, "Full characterization of act-and-wait control for first order unstable lag processes,"Journal of Vibration and Control (in press).
18.
Israelsson, D. and Johnsson, A., 1967, "A theory for circumnutations in Helianthus annuus ," Physiologia Plantarum20, 957-976.
19.
Kalmár-Nagy, T., 2005, "A new look at the stability analysis of delay-differential equations," in Proceedings of the DETC, Long Beach, CA, September 24-28.
20.
Kalmár-Nagy, T., Pratt, J.R., Davies, M.A., and Kennedy, M.D., 1999, "Experimental and analytical investigation of the subcritical instability in turning," in Proceedings of the 1999 ASME Design Engineering Technical Conferences, 17th ASME Biennial Conference on Mechanical Vibration and Noise, Las Vegas, NV, DETC99/VIB-8060.
21.
Kalmár-Nagy, T., Stépán G., and Moon, F.C., 2001, "Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations," Nonlinear Dynamics26, 121-142.
22.
Kim, Y.B., Noah S.T., and Choi, Y.S., 1991, "Periodic response of multi-disk rotors with bearing clearances ," Journal of Sound and Vibrations144(3), 381.
23.
Krasnolselskii, M.A., 1984, "On the problem of forced oscillation in a nonlinear control systems," Engineering Cybernetics22, 92-99.
24.
Kuang, Y., 1993, Delay Differential Equations with Applications to Population Dynamics, Academic Press, New York .
25.
Liu, L. and Dowell, E.H., 2005, "Harmonic balance approach for an airfoil with a freeplay control surface," AIAA Journal43(4), 802.
26.
Liu, L., Thomas, J.P., Dowell, E.H., Attar, P., and Hall, K.C., 2006, "A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator ," Journal of Computational Physics215(1), 298-320.
27.
MacDonald, N., 1989, Biological Delay Systems: Linear Stability Theory , Cambridge University Press, New York.
28.
MacDonald, N., 1995, "Harmonic balance in delay-differential equations ," Journal of Sound and Vibration186(4), 649-656.
29.
Mann, B.P. and Patel, B.R., 2010, "Stability of delay equations written as state space models,"Journal of Vibration and Control (in press).
30.
McMullen, M., Jameson, A., and Alonso, J.J., 2001, "Acceleration of convergence to a periodic steady state in turbomachinery flows," AIAA Paper 2001-0152.
31.
Mickens, R.E., 1996, Oscillations in Planar Dynamic Systems (Advances in Mathematics for Applied Sciences, Vol. 37), World Scientific , River Edge, New Jersey.
32.
Morris, H.C., 1976, "A perturbative approach to periodic solutions of delay differential equations," IMA Journal of Applied Mathematics18(1), 15-24.
33.
Ohlsson, H., 2006, "Mathematical analysis of a biological clock model ," Masters dissertation, Linkopings University, Sweden.
34.
Olgac, N. and Sipahi, R., 2002, "An exact method for the stability analysis of time-delayed LTI systems," IEEE Transactions on Automatic Control47(5), 793-797.
35.
Pieroux, D., Erneux, T., Luzyanina, T., and Engelborghs, K., 2001, "Interacting pairs of periodic solutions lead to tori in lasers subject to delayed feedback," Physical Review E63(3), 36211.
36.
Pyragas, K., 1992, "Continuous control of chaos by self-controlling feedback ," Physical Review Letters78(2), 421- 428.
37.
Ragulskis, M., Fedaravicius, A., and Ragulskis K., 2006, "Harmonic balance method for FEM analysis of fluid flow in a vibrating pipe," Communications in Numerical Methods in Engineering22(5), 347-356.
38.
Saupe, D., 1983, "Global bifurcation of periodic solutions of some autonomous differential delay equations," Applied Mathematics and Computing13, 185-211.
39.
Shayer, L. and Campbell, S.A., 2000, "Stability, bifurcation, and multi-stability in a system of two coupled neurons with multiple time delays," SIAM Journal of Applied Mathematics61, 673-700.
40.
Stépán, G., 1989, Retarded Dynamical Systems: Stability and Characteristic Functions (Pitman Research Notes in Mathematics, Vol. 210), Longman Scientific and Technical, Harlow.
41.
Stépán, G., 1997, "Delay-differential equation models for machine tool chatter ," in F. C. Moon (ed.), Dynamics and Chaos in Manufacturing Processes, Wiley, New York, pp. 165-191.
42.
Stépán, G., 2000, "Modeling nonlinear regenerative effects in metal cutting ," Philosophical Transactions of the Royal Society of London A359, 739-757.
43.
Szalai, R., Stépán, G., and Hogan, S.J., 2006, "Continuation of bifurcations in periodic delay-differential equations using characteristic matrices," SIAM Journal on Scientific Computing28(4), 1301-1317.
44.
Thomas, J.P., Dowell, E.H., and Hall, K.C., 2002a, "Nonlinear inviscid aerodynamic effects on transonic divergence, flutter, and limit-cycle oscillations" AIAA Journal40(4), 638-646.
45.
Thomas, J.P., Dowell, E.H., and Hall, K.C., 2002b, "Modeling viscous transonic limit cycle oscillation behavior using a harmonic balance approach," AIAA Paper 2002-1414.
46.
Thomas, J.P., Hall, K.C., and Dowell, E.H., 2003, "A harmonic balance approach for modeling nonlinear aeroelastic bahvior of wings in transonic flow," AIAA Paper 2003-1924.
47.
Verdugo, A. and Rand, R., 2008, "Hopf bifurcation in a DDE model of gene expression ," Communications in Nonlinear Science and Numerical Simulation13, 235-242.
48.
Wahi, P. and Chatterjee, A., 2005, "Galerkin projections for delay differential equations ," Journal of Dynamic Systems, Measurement, and Control127(1), 80-87.
49.
Wang, H. and Hu, H.Y., 2003, "Remarks on the perturbation methods in solving the second order delay differential equations," Nonlinear Dynamics33, 379-398.
50.
Wu, D.M. and Wang, Z., 2006, "A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach ," Computer Physics Communications174(6), 447-463.
51.
Yi, S., Ulsoy, A.G., and Nelson, P.W., 2006, "Analysis of systems of linear delay differential equations using the matrix Lambert function and the Laplace transformation," in Proceedings of CDC 2006, San Diego, CA.
