An energy analysis of the local dynamics of a delayed oscillator near a hopf bifurcation

Nonlinear Dynamics

Volumn 46, Issue 1-2, 2006, Pages 149-159

  Wang, Z H ^a   Hu, H Y ^a  

^a NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS   (China)

Author keywords

Energy analysis; Hopf bifurcation; Self excitation; Stability; Time delay

Indexed keywords

BIFURCATION (MATHEMATICS); COMPUTATIONAL METHODS; DELAY CIRCUITS; FUNCTIONS; LYAPUNOV METHODS; NUMERICAL ANALYSIS;

ENERGY ANALYSIS; HOPF BIFURCATION; SELF-EXCITATION; TIME DELAY;

OSCILLATORS (ELECTRONIC);

EID: 33748539067     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-006-9020-5     Document Type: Article

References (18)

1. Hu, H. Y. and Wang, Z.H., Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer-Verlag, Heidelberg, 2002.
2. Niculescue, S.-L, Delay Effects on Stability: A Robust Approach, Springer-Verlag, London, 2001.
3. Qin, Y. X., Liu, Y. Q., Wang, L., and Zhen, Z.X., Stability of Dynamic Systems with Delays, Science Press, Beijing, 1989.
4. Campbell, S. A., Belair, J., Ohira, T., and Milton, J., 'Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback', Chaos 5, 1995, 640-645.
5. Stepan, G., 'Modelling nonlinear regenerative effects in metal cutting', Philosophical Transactions of the Royal Society of London 359, 2001, 739-757.
6. Xu, J. and Lu, Q. S., 'Hopf bifurcation of time-delay Lienard equations', International Journal of Bifurcation and Chaos 9, 1999, 939-951.
7. Das, S. L. and Chatterjee, A., 'Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations', Nonlinear Dynamics 30, 2002, 323-335.
8. Hu, H. Y., Dowell, E. H., and Virgin, L. N., 'Resonances of a harmonically forced Duffing oscillator with time delay state feedback', Nonlinear Dynamics 15, 1998, 311-327.
9. Wang, H. L., Hu, H. Y., and Wang, Z. H., 'Global dynamics of Duffing oscillator with delayed displacement feedback', International Journal of Bifurcation and Chaos 14, 2004, 2753-2775.
10. Hale, J. K. and Kocak, H., Dynamics and Bifurcations, Springer-Verlag, New York, 1991.
11. Hale, J. K. and Lunel, S. M. V., Introduction to Functional Diffrential Equations, Springer-Verlag, New York, 1993.
12. Hale, J. K., 'Averaging methods for differential equations with retarded arguments and a small parameter', Journal of Differential Equations 2, 1966, 57-73.
13. Sato, K. Yamamoto, S., Okimura, T., and Watanabe, S., 'Dynamic motion of a nonlinear mechanical system with time delay', Transactions of JSME, Series C, 61, 1995, 101-106.
14. Yoshitake, Y., Inoue, J., and Sueoka, A., Vibration of a forced self-excited system with time delay, Transactions of JSME, Series C, 49, 1984, 298-305.
15. Minorsky, N., Nonlinear Oscillations, D. Van Nostrand, Princeton, 1962.
16. Wang, Z. H., Hu, H. Y., and Wang, H. L., 'Robust stabilization to nonlinear delayed systems via delayed state feedback: The averaging method', Journal of Sound and Vibration 279, 2005, 937-953.
17. Wang, Z. H. and Hu, H. Y., An energy analysis of nonlinear oscillators with time delays, International Journal of Bifurcation and Chaos 2006, 16(7): accepted for publication
18. Wang, Z. H. and Hu, H. Y., Pseudo-energy analysis of scalar nonlinear dynamics with time delays near Hopf bifurcation, International Journal of Bifurcation and Chaos, 2005