A large class of strongly nonlinear conservative oscillators subject to a delayed feedback are modeled mathematically by second-order delay differential equations. Recent applications include the control of crane oscillations and lasers subject to optoelectronic feedback. We apply the method of averaging in the case of weak damping and weak feedback and determine the bifurcation diagram of the limit-cycle solutions. We find that the coexistence of a stable equilibrium with one or several stable periodic solutions is unavoidable if the delay is sufficiently large.
Abramowitz, M. and Stegun, I.A., 1972, Handbook of Mathematical Functions, 9th edn, Dover, New York.
2.
Bourland, F.J. and Haberman, R., 1988, "The modulated phase shift for strongly nonlinear, slowly varying, and weakly damped oscillators," SIAM Journal of Applied Mathematics48, 737-748.
3.
Bourland, F.J., Haberman, R., and Kath, W.L., 1991, "Averaging methods for the phase shift of arbitrarily perturbed strongly nonlinear oscillators with an application to capture," SIAM Journal of Applied Mathematics51, 1150-1167.
4.
Drazin, P.G., 1992, Nonlinear Systems (Cambridge Texts in Applied Mathematics) , Cambridge University Press, Cambridge .
5.
Erneux, T., 2000, "Multiple time scale analysis of lasers," in Fundamental Issues of Nonlinear Dynamics (AIP Conf. Proc., 548), Krauskopf, B. and Lenstra, D. (eds). American Institute of Physics, New York, p. 54-65.
6.
Erneux, T. and Kalmár-Nagy, T., 2007, "Nonlinear stability of a delayed feedback controlled container crane," Journal of Vibration and Control13, 603-616.
7.
Henry, R.J., Masoud, Z.N., Nayfeh, A.H., and Mook, D.T., 2001, "Cargo pendulation reduction on ship-mounted cranes via boom-LU angle actuation," Journal of Vibration and Control7, 1253-1264.
8.
Kevorkian, J. and Cole, J.D., 1981, Perturbation Methods in Applied Mathematics (Applied Mathematical Sciences, 34), Springer, New York .
9.
Kevorkian, J. and Cole, J.D., 1996, Multiple Scale and Singular Perturbation Methods (Applied Mathematical Sciences, 114), Springer, New York.
10.
Masoud, Z.N., Nayfeh, A.H., and Al-Mousa, A., 2003, "Delayed position-feedback controller for the reduction of payload pendulations of rotary cranes," Journal of Vibration and Control9, 257-277.
11.
Masoud, Z.N., Nayfeh, A.H., and Nayfeh, N.A., 2005, "Sway reduction on quay-side container cranes using delayed feedback controller: simulations and experiments," Journal of Vibration and Control11, 1103- 1122.
12.
Nayfeh, N., Masoud, Z., and Baumann, W., 2005, "A comparison of three feedback controllers for container cranes," in Proceedings of IDETC/CIE 2005, ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference , Long Beach, CA, September 24-28.
13.
Pieroux, D. and Erneux, T., 1996, "Strongly pulsating lasers with delay," Physics Review A53, 2765-2771.
14.
Pieroux, D., Erneux, T., and Otsuka, K., 1994, "Minimal model of a class-B laser with delayed feedback: cascading branching of periodic solutions and period-doubling bifurcation ," Physics Review A50, 1822-1829.
15.
Pieroux, D., Erneux, T., Gavrielides, A., and Kovanis, V., 2000, "Hopf bifurcation subject to a large delay in a laser system," SIAM Journal of Applied Mathematics61, 966-982.
16.
Pyragas, K., 1992, "Continuous control of chaos by self-controlling feedback ," Physics Letters A170, 421-428.
17.
Verhulst, F., 2005, Methods and Applications of Singular Perturbations (Texts in Applied Mathematics, 50), Springer, New York.
18.
Weiss, C.O. and Vilaseca, R., 1991, Dynamics of Lasers, VCH, Weinheim (FRG).
19.
Xia, G.-Q., Chan, S.-C., and Liu, J.-M., 2007, "Multistability in a semiconductor laser with optoelectronic feedback," Optics Express15, 572-576.
