Analytical approximations for the periodic motion of the Duffing system with delayed feedback

Numerical Algorithms

Volumn 56, Issue 4, 2011, Pages 561-576

  You, Xiangcheng ^a   Xu, Hang ^a  

^a Shanghai Jiao Tong University   (China)

Author keywords

Delayed differential equation; Homotopy analysis method; Periodic motion

Indexed keywords

EID: 79952282378     PISSN: 10171398     EISSN: 15729265     Source Type: Journal    
DOI: 10.1007/s11075-010-9404-y     Document Type: Article

References (52)

1. Stépán, G.: Modeling nonlinear regenerative effects in metal cutting. Philos. Trans. R. Soc. Lond., A 359, 739-757 (2000).
2. Perioux, D., Erneux, T., Gavrielides, A., Kovanis, V.: Hopf bifurcation subject to a large delay in a laser system. SIAM J. Appl. Math. 61, 966-982 (2000).
3. Shayer, L., Campbell, S. A.: Stability, bifurcation, and multi-stability in a system of two coupled neurones with multiple time delays. SIAM J. Appl. Math. 61, 673-700 (2000).
4. MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University, New York (1989).
5. Beuter, A., Belair, J., Labrie, C.: Feedback and delays in neurological diseases: a modelling study using dynamical systems. Bull. Math. Biol. 55, 525-541 (1993).
6. Qin, Y. X., Liu, Y. Q., Wang, L., Zheng, Z. X.: Stability of Dynamic Systems with Delays. Science, Beijing (1989). In Chinese.
7. Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. Longman Scientific and Technical, Essex (1989).
8. Kuang, Y.: Delay Differential Equations with Applications to Population Dynamics. Academic, New York (1993).
9. Hu, H. Y., Wang, Z. H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, Berlin (2002).
10. Wang, H. L.: Dynamics of Duffing Systems Under Delayed State Feedback Control. PhD thesis, Nanjing University of Aeronautics and Astronautics (2003).
11. Atay, F. M.: van der pol's oscillator under delayed feedback. J. Sound Vibr. 218, 333-339 (1998).
12. Hu, H. Y.: Abundant dynamic features of a nonlinear system under delayed feedback control. In: Proc. 6th Asia-Pacific Vibr. Confer., vol. 1, pp. 11-15 (2001).
13. de Oliveira, J. C. F.: Oscillations in a van der Pol equation with delayed argument. J. Math. Anal. Appl. 275, 789-803 (2002).
14. Shang, H. L., Xu, J.: Multiple periodic solutions in delayed duffing equation. J. Taiyuan Univ. Tech. 36, 749-751 (2005).
15. Shang, H. L., Xu, J.: Multiple periodic solutions in Liénard oscillator with delayed position feedbacks. J. Tongji Univ. 36, 962-966 (2008).
16. Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Rev. Lett. 78, 421-428 (1992).
17. Yamamoto, S., Hino, T., Ushio, T.: Delayed feedback control with a minimal-order observer for stabilization of chaotic discrete-time systems. Int. J. Bifurc. Chaos 12, 1047-1055 (2002).
18. Bélair, J., Campbell, S. A.: Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J. Appl. Math. 54, 1402-1424 (1994).
19. Hu, H. Y., Wang, Z. H.: Review on nonlinear dynamics systems involving time delays. Adv. Mech. 29, 501-512 (1999).
20. Reddy, D. V. R., Sen, A., Johnston, G. L.: Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks. Physica, D 44, 335-357 (2000).
21. Xu, J., Chung, K. W.: Effects of time delayed position feedback on a van der Pol-Duffing oscillator. Physica, D 180, 17-39 (2003).
22. Wang, H. L., Hu, H. Y.: Remarks on the perturbation methods in solving the second-order delay differential equations. Physica, D 33, 379-398 (2003).
23. Dai, H. J., Xu, J.: Effects of time delay on periodic motions in nonlinear system with parametric excitation. Chin. Quart. Mech. 25, 367-374 (2004).
24. Xu, R., Chaplain, M. A. J., Davidson, F. A.: Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments. Nonlinear Anal.: Real World Appl. 5, 183-206 (2004).
25. Wang, H. L., Hu, H. Y.: A note on the method of multiple scales. J. Dyn. Control 2, 10-13 (2004).
26. Hu, H. Y.: Global dynamics of a Duffing system with delayed velocity feedback. In: Proc. IUTAM Symp. Chaotic Dyn. Control. Syst. Process Mech., pp. 335-344. Springer, Dordrecht (2004).
27. Wang, H. L., Hu, H. Y., Wang, Z. H.: Global dynamics of a Duffing oscillator with delayed displacement feedback. Int. J. Bifurc. Chaos 14, 2753-2775 (2004).
28. Hu, H. Y., Wang, Z. H.: Singular perturbation methods for nonlinear dynamic systems with time delays. Chaos, Solitons Fractals 40, 13-27 (2009).
29. Hassard, B. D., Kazarinoff, N. D., Wan, Y. H.: Counting roots of the characteristic equation for linear-delay differential systems. J. Differ. Equ. 136, 222-235 (1997).
30. Hu, H. Y., Dowell, E. H., Virgin, L. N.: Resonances of a harmonically forced duffing oscillator with time delay state feedback. Nonlinear Dyn. 15, 311-323 (1998).
31. Wang, Z. H., Hu, H. Y.: Delay-independent stability of retarded dynamic systems of multiple degrees of freedom. J. Sound Vibr. 226, 57-81 (1999).
32. Wang, Z. H., Hu, H. Y.: Stability switches of time-delayed dynamic systems with unknown parameter. J. Sound Vibr. 233, 215-233 (2000).
33. Yu, P., Yuan, Y., Xu, J.: Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback. Comm. Nonlinear Sci. Numer. Simulat. 7, 69-91 (2002).
34. Wang, H. L., Hu, H. Y.: Bifurcation analysis of a delayed dynamic system via method of multiple scales and shooting technique. Int. J. Bifurc. Chaos 15, 425-450 (2005).
35. Wang, Z. H., Hu, H. Y.: Stabilization of vibration systems via delayed state difference feedback. J. Sound Vibr. 29, 117-129 (2006).
36. Wang, Z. H., Hu, H. Y.: An energy analysis of the local dynamics of a delayed oscillator near a Hopf bifurcation. Nonlinear Dyn. 46, 149-159 (2006).
37. Khan, H., Liao, S. J., Mohapatra, R. N., Vajravelu, K.: An analytical solution for a nonlinear time-delay model in biology. Comm. Nonlinear Sci. Numer. Simulat. 14, 3141-3148 (2009).
38. Liao, S. J.: The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD thesis, Shanghai Jiao Tong University (1992).
39. Liao, S. J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC, Boca Raton (2003).
40. Liao, S. J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499-513 (2004).
41. Abbasbandy, S.: The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett., A 360, 109-113 (2006).
42. Zhu, S. P.: An exact and explicit solution for the valuation of American put options. Quant. Financ. 6, 229-242 (2006).
43. Yamashita, M., Yabushita, K., Tsuboi, K.: An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys., A 40, 8403-8416 (2007).
44. Tao, L., Song, H., Chakrabarti, S.: Nonlinear progressive waves in water of finite depth-an analytic approximation. Coast. Eng. 54, 825-834 (2007).
45. Liao, S. J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297-355 (2007).
46. Cheng, J., Liao, S. J.: Analytical approximations for nonlinear dynamic system with multiple limit cycles. Chin. J. Theor. Appl. Mech. 39, 715-720 (2007).
47. Wu, Y., Cheung, K. F.: Explicit solution to the exact Riemann problems and application in nonlinear shallow water equations. Int. J. Numer. Methods Fluids 57, 1649-1668 (2008).
48. Liao, S. J.: Notes on the homotopy analysis method: some definitions and theorems. Comm. Nonlinear Sci. Numer. Simulat. 14, 983-997 (2009).
49. Marinca, V., Herisanu, N.: Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int. Commun. Heat Mass Transf. 35, 710-715 (2008).
50. Marinca, V., Hersanu, N., Bota, C., Marinca, B.: An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Appl. Math. Lett. 22, 245-251 (2009).
51. Niu, Z., Wang, C.: A one-step optional homotopy analysis method for nonlinear differential equations. Comm. Nonlinear Sci. Numer. Simulat. 15, 2026-2036 (2010).
52. Liao, S. J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Comm. Nonlinear Sci. Numer. Simulat. 15, 2003-2016 (2010).