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Global dynamics of a Duffing oscillator with delayed displacement feedback. (English) Zbl 1075.34077
The present paper gives a systematic study of the abundant dynamics, especially the stability switches and local bifurcations, of a Duffing oscillator with delayed displacement feedback. The formulas for computing the critical values of time delay for the stability switches are given for different combinations of the system parameters and the time delay, the global diagrams of local bifurcations for periodic motions with respect to the time delay are obtained for other given system parameters, and then the local bifurcation diagrams including the saddle-node bifurcation and pitchfork bifurcation are presented. In addition, a novel feature of “periodicity in delay” is observed in the global diagrams of local bifurcations and is used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. Moreover, the stability of the periodic motions is discussed from numerical simulations as well as from the viewpoint of the basin of attraction . Furthermore, a conventional Poincaré section technique is used to reveal interesting dynamical structures of attractors for different time delays and the typical feature of tori bifurcation sequence to chaos. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one. It has been found that the dynamical structures on the two kinds of Poincaré section are topologically rotational-symmetric.
Reviewer: Jin Zhou (Tianjin)

MSC:
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
93D15 Stabilization of systems by feedback
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References:
[1] Bünner M. J., Phys. Rev. 54 pp R3082–
[2] Bünner M. J., Phys. Rev. 56 pp 5083–
[3] Campbell S. A., Canad. Appl. Math. Quart. 7 pp 217–
[4] Casal A., IEEE Trans. Auto. Cont. 25 pp 967– · Zbl 0437.34058 · doi:10.1109/TAC.1980.1102450
[5] Diekmann O., Delay Equations, Functional, Complex, and Nonlinear Analysis (1995) · Zbl 0826.34002
[6] DOI: 10.1006/jdeq.1995.1144 · Zbl 0836.34068 · doi:10.1006/jdeq.1995.1144
[7] DOI: 10.1016/0022-0396(72)90064-2 · Zbl 0231.34063 · doi:10.1016/0022-0396(72)90064-2
[8] DOI: 10.1007/978-1-4612-9892-2 · doi:10.1007/978-1-4612-9892-2
[9] DOI: 10.1115/1.3438305 · doi:10.1115/1.3438305
[10] Hassard B. D., Theory and Applications of Hopf Bifurcation (1981) · Zbl 0474.34002
[11] DOI: 10.1006/jdeq.1996.3127 · Zbl 0872.34051 · doi:10.1006/jdeq.1996.3127
[12] DOI: 10.1016/0022-0396(83)90037-2 · Zbl 0477.93040 · doi:10.1016/0022-0396(83)90037-2
[13] DOI: 10.1023/A:1008278526811 · Zbl 0906.34052 · doi:10.1023/A:1008278526811
[14] H. Y. Hu, Proc. 6th Asia-Pacific Vibration Conf. 1 (2001) pp. 11–15.
[15] DOI: 10.1007/978-3-662-05030-9 · doi:10.1007/978-3-662-05030-9
[16] DOI: 10.1142/S0218127499000031 · Zbl 0965.70002 · doi:10.1142/S0218127499000031
[17] Kuang Y., Delay Differential Equations with Application to Population Dynamics (1993) · Zbl 0777.34002
[18] DOI: 10.1142/S0218127401003425 · Zbl 1091.70502 · doi:10.1142/S0218127401003425
[19] DOI: 10.1115/1.3153746 · doi:10.1115/1.3153746
[20] DOI: 10.1002/9783527617500 · doi:10.1002/9783527617500
[21] DOI: 10.1093/imamat/18.1.15 · Zbl 0346.34057 · doi:10.1093/imamat/18.1.15
[22] A. Nayfeh, C. Chin and J. Pratt, Dynamics and Chaos in Manufacturing Processes, ed. F. C. Moon (John Wiley, 1997) pp. 193–213.
[23] DOI: 10.1016/S0022-460X(87)80068-8 · Zbl 1235.74108 · doi:10.1016/S0022-460X(87)80068-8
[24] Qin Y. X., Stability of Dynamic Systems with Delay (1989)
[25] DOI: 10.1137/S0036139998344015 · Zbl 0992.92013 · doi:10.1137/S0036139998344015
[26] Stépán G., Retarded Dynamical Systems: Stability and Characteristic Functions (1989) · Zbl 0686.34044
[27] Stépán G., Nonlin. Dyn. 8 pp 513–
[28] Stépán G., Phil. Trans. R. Soc. Lond. 359 pp 739– · Zbl 1169.74431 · doi:10.1098/rsta.2000.0753
[29] Wang H. L., Acta Mech. Sin. 4
[30] DOI: 10.1006/jsvi.1999.2282 · Zbl 1235.70002 · doi:10.1006/jsvi.1999.2282
[31] DOI: 10.1006/jsvi.1999.2817 · Zbl 1237.93159 · doi:10.1006/jsvi.1999.2817
[32] DOI: 10.1023/A:1008270815516 · Zbl 0914.70020 · doi:10.1023/A:1008270815516
[33] DOI: 10.1016/S1007-5704(02)00007-2 · Zbl 1010.34070 · doi:10.1016/S1007-5704(02)00007-2
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