Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback

Acta Mechanica Sinica/Lixue Xuebao

Volumn 20, Issue 4, 2004, Pages 426-434

  Wang, Huailei ^a   Wang, Zaihua ^b   Hu, Haiyan ^a  

^a NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS   (China)

^b PLA UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

Author keywords

Delay differential equation; Fredholm alternative; Stability switches; Supercritical Hopf bifurcation

Indexed keywords

BIFURCATION (MATHEMATICS); DIFFERENTIAL EQUATIONS; NONLINEAR EQUATIONS; OSCILLATORS (MECHANICAL); STATE FEEDBACK;

DELAYED VELOCITY FEEDBACK; FREDHOLM ALTERNATIVE; PERIODIC MOTIONS; STABILITY SWITCHES; TIME DELAY;

DYNAMIC MECHANICAL ANALYSIS;

EID: 7944221027     PISSN: 05677718     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf02489381     Document Type: Article

References (20)

1. Hale JK. Theory of Functional Differential Equations. New York: Springer-Verlag, 1977
2. Qin YX, et al. Stability of Dynamic Systems with Delays. Beijing: Science Press, 1989
3. Kuang Y. Delay Differential Equations with Applications to Population Dynamics, New York: Academic Press, 1993
4. Diekmann O, van Gils SA, Verduyn Lunel SM, et al. Delay Equations, Functional, Complex, and Nonlinear Analysis. New York: Springer-Verlag, 1995
5. Hu HY, Wang ZH. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Heidelberg: Springer-Verlag, 2002
6. van der Heiden U, Walther HO. Existence of chaos in control system with delayed feedback. Journal of Differential Equations, 1983, 47: 273-295
7. Shayer L, Campbell SA. Stability, bifurcation, and multi-stability in a system of two coupled neurons with multiple time delays. SIAM Journal of Applied Mathematics, 2000, 61: 673-700
8. Campbell SA, Belair T, et al. Limit cycles, tori, and complex dynamics in a second-order differential equations with delayed negative feedback. Journal of Dynamics and Differential Equations, 1995, 7: 213-236
9. Casal A, Freedman MA. Poincare-Lindstedt approach to bifurcation problems for differential-delay equations. IEEE Transactions on Automatic Control, 1980, AC-25: 967-973
10. Plaut RH, Hsieh JC. Nonlinear structural vibrations involving a time delay in damping. Journal of Sound and Vibration, 1987, 117: 497-510
11. Hu HY, Dowell EH, Virgin LN. Resonances of a harmonically forced Duffing oscillator with time delay state feedback. Nonlinear Dynamics, 1998, 15: 311-327
12. Moiola JL, Chen G. Hopf bifurcations in time-delayed nonlinear feedback control systems. In: Michael A. Peshkin ed. Proceedings of the 34th Conference on Decision and Control, New Orleans, New York: IEEE. 1995. 942-948
13. Xu J, Lu QS. Hopf bifurcation of time-delay Lienand equations. International Journal of Bifurcation and Chaos, 1999, 9: 939-951
14. Xu J, Cheung KW. Effects of time delayed position feedback on a van der Pol-Duffing oscillator. Physica D, 2003, 3096: 1-23
15. Stepan G. Modelling nonlinear regenerative effects in metal cutting. Philosophical Transactions of the Royal Society of London, 2001, A359: 739-757
16. Liao XF, Chen GR. Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two time delays. International Journal of Bifurcation and Chaos, 2001, 11: 2105-2121
17. Faria F, Magalhaes LT. Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. Journal of Differential Equations, 1995, 122: 181-200
18. Wang ZH, Hu HY. Stability switches of time-delayed dynamic systems with unknown parameters. Journal of Sound and Vibration, 2000, 233: 215-233
19. Iooss G, Joseph DD. Elementary Stability and Bifurcation Theory. New York: Springer-Verlag, 1980
20. Nayfeh AH, Chin CM, Pratt J. Perturbation methods in nonlinear dynamics: applications to machine dynamics. Journal of Manufacturing Science and Engineering, 1997, 119: 485-493