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With increasingly strict requirements for control speed and system performance, the unavoidable time delay becomes a serious problem. Fractional-order feedback is constantly adopted in control engineering due to its advantages, such as robustness, strong de-noising ability and better control performance. In this paper, the dynamical characteristics of an autonomous Duffing oscillator under fractional-order feedback coupling with time delay are investigated. At first, the first-order approximate analytical solution is obtained by the averaging method. The equivalent stiffness and equivalent damping coefficients are defined by the feedback coefficient, fractional order and time delay. It is found that the fractional-order feedback coupling with time delay has the functions of both delayed velocity feedback and delayed displacement feedback simultaneously. Then, the comparison between the analytical solution and the numerical one verifies the correctness and satisfactory precision of the approximately analytical solution under three parameter conditions respectively. The effects of the feedback coefficient, fractional order and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed, including the locations of bifurcation points, the stabilities of the periodic solutions, the existence ranges of the periodic solutions, the stability of zero solution and the stability switch times. It is found that the increase of fractional order could make the delay-amplitude curves of periodic solutions shift rightwards, but the stabilities of the periodic solutions and the stability switch times of zero solution cannot be changed. The decrease of the feedback coefficient makes the amplitudes and ranges of the periodic solutions become larger, and induces the stability switch times of zero solution to decrease, but the stabilities of the periodic solutions keep unchanged. The sign of the nonlinear stiffness coefficient determines the stabilities and the bending directions of delay-amplitude curves of periodic solutions, but the bifurcation points, the stability of zero solution and the stability switch times are not changed. It could be concluded that the primary system parameters have important influences on the dynamical behavior of Duffing oscillator, and the results are very helpful to design, analyze or control this kind of system. The analysis procedure and conclusions could provide a reference for the study on the similar fractional-order dynamic systems with time delays.
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Keywords:
- fractional-order derivative /
- Duffing oscillator /
- time delay /
- averaging method
[1] Gorenflo R, Abdel-Rehim E A 2007 J. Comput. Appl. Math. 205 871
[2] Jumarie G 2006 Comput. Math. Appl. 51 1367
[3] Ishteva M, Scherer R, Boyadjiev L 2005 Math. Sci. Res. J. 2005 9 161
[4] Agnieszka B M, Delfim F M T 2011 Fract. Calc. Appl. Anal. 14 523
[5] Leung A Y T, Guo Z J, Yang H X 2012 J. Sound Vib. 331 1115
[6] Yang J H, Zhu H 2013 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2013 物理学报 62 024501]
[7] Zhou Y, Ionescu C, Machado J A T 2015 Nonlinear Dyn. 80 1661
[8] Wang Z H, Hu H Y 2009 Sci. China G: Phys. Mech. Astron. 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑: 物理学 力学 天文学 39 1495]
[9] Wang Z H, Du M L 2011 Shock Vib. 18 257
[10] Shen Y J, Wei P, Yang S P 2014 Nonlinear Dyn. 77 1629
[11] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 150503 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 150503]
[12] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 110505]
[13] Shen Y J, Yang S P, Xing H J, Gao G S 2012 Commun. Nolinear Sci. 17 3092
[14] Shen Y J, Yang S P, Xing H J, Ma H X 2012 Int. J. Nonlin. Mech. 47 975
[15] Li C P, Deng W H 2007 Appl. Math. Comput. 187 777
[16] Deng W H, Li C P 2008 Phys. Lett. 372 401
[17] Li Q D, Chen S, Zhou P 2011 Chin. Phys. B 20 010502
[18] Chen L C, Hu F, Zhu W Q 2013 Fract. Calc. Appl. Anal. 05 189
[19] Wahi P, Chatterjee A 2004 Nonlinear Dyn. 38 3
[20] Yin H, Chen N 2012 Chin. J. Comput. Mech. 29 966 (in Chinese) [银花, 陈宁 2012 计算力学学报 29 966]
[21] Xu Y, Li Y G, Liu D 2013 Nonlinear Dyn. 74 745
[22] Zhang R X, Yang S P 2009 Chin. Phys. B 18 3295
[23] Zhang R R, Xu W, Yang G D 2015 Chin. Phys. B 24 020204
[24] Hale J K, Lunel S M V 1993 Introduction to Functional Differential Equations (New York: Springer-Verlag) p6
[25] Hu H Y, Wang Z H 2002 Dynamics of Controlled Mechanical Systems with Delayed Feedback (Berlin: Springer) p213
[26] Wang Z H, Hu H Y 2000 J. Sound Vib. 233 215
[27] Wang H L, Hu H Y 2003 Nonlinear Dyn. 33 379
[28] Shi M, Wang Z H 2011 Automatica 47 2001
[29] Babakhani A, Baleanu D, Khanbabaie R 2012 Nonlinear Dyn. 69 721
[30] elik V, Demir Y 2014 Signal Image Video P. 8 65
[31] Petras I 2011 Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation (Beijing: Higher Education Press) p19
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[1] Gorenflo R, Abdel-Rehim E A 2007 J. Comput. Appl. Math. 205 871
[2] Jumarie G 2006 Comput. Math. Appl. 51 1367
[3] Ishteva M, Scherer R, Boyadjiev L 2005 Math. Sci. Res. J. 2005 9 161
[4] Agnieszka B M, Delfim F M T 2011 Fract. Calc. Appl. Anal. 14 523
[5] Leung A Y T, Guo Z J, Yang H X 2012 J. Sound Vib. 331 1115
[6] Yang J H, Zhu H 2013 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2013 物理学报 62 024501]
[7] Zhou Y, Ionescu C, Machado J A T 2015 Nonlinear Dyn. 80 1661
[8] Wang Z H, Hu H Y 2009 Sci. China G: Phys. Mech. Astron. 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑: 物理学 力学 天文学 39 1495]
[9] Wang Z H, Du M L 2011 Shock Vib. 18 257
[10] Shen Y J, Wei P, Yang S P 2014 Nonlinear Dyn. 77 1629
[11] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 150503 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 150503]
[12] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 110505]
[13] Shen Y J, Yang S P, Xing H J, Gao G S 2012 Commun. Nolinear Sci. 17 3092
[14] Shen Y J, Yang S P, Xing H J, Ma H X 2012 Int. J. Nonlin. Mech. 47 975
[15] Li C P, Deng W H 2007 Appl. Math. Comput. 187 777
[16] Deng W H, Li C P 2008 Phys. Lett. 372 401
[17] Li Q D, Chen S, Zhou P 2011 Chin. Phys. B 20 010502
[18] Chen L C, Hu F, Zhu W Q 2013 Fract. Calc. Appl. Anal. 05 189
[19] Wahi P, Chatterjee A 2004 Nonlinear Dyn. 38 3
[20] Yin H, Chen N 2012 Chin. J. Comput. Mech. 29 966 (in Chinese) [银花, 陈宁 2012 计算力学学报 29 966]
[21] Xu Y, Li Y G, Liu D 2013 Nonlinear Dyn. 74 745
[22] Zhang R X, Yang S P 2009 Chin. Phys. B 18 3295
[23] Zhang R R, Xu W, Yang G D 2015 Chin. Phys. B 24 020204
[24] Hale J K, Lunel S M V 1993 Introduction to Functional Differential Equations (New York: Springer-Verlag) p6
[25] Hu H Y, Wang Z H 2002 Dynamics of Controlled Mechanical Systems with Delayed Feedback (Berlin: Springer) p213
[26] Wang Z H, Hu H Y 2000 J. Sound Vib. 233 215
[27] Wang H L, Hu H Y 2003 Nonlinear Dyn. 33 379
[28] Shi M, Wang Z H 2011 Automatica 47 2001
[29] Babakhani A, Baleanu D, Khanbabaie R 2012 Nonlinear Dyn. 69 721
[30] elik V, Demir Y 2014 Signal Image Video P. 8 65
[31] Petras I 2011 Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation (Beijing: Higher Education Press) p19
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