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A forced Holmes-Duffing system is considered in this paper. A delayed velocity feedback is opplied to the system. The erosion of safe basin, caused by the periodic excitation, and the effects of delayed velocity feedback on the controlling of the fractal eroded safe basin are investigated. The conditions of fractal erosion of the basin boundary are obtained by the Melnikov method. Then considering the time delay as a variable parameter, the evolutions of safe basin with time delay are presented numerically by the 4th Runge-Kutta and the point-to-point mapping method. It is found that the increase of time delay can enhance the threshold of the fractal erosion of the basin boundary under a weak and positive feedback gain so as to reduce the basin erosion. These imply that the delayed velocity feedback can control the basin erosion of the system effectively.
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Keywords:
- erosion of safe basin /
- fractal /
- Melnikov function /
- delayed feedback control
[1] Thompson J M T, Rainey F C T, Soliman M S 1995 Philosophical Transactions of the Royal Society 332 149
[2] Soliman M S 1995 J. Sound Vib. 182 618
[3] Gu J Y, Miao Z H 2005 Journal of Jiangsu University of Science and Technology (Natural Science Edition) 19 6 (in Chinese) [谷家扬, 缪振华 2005 江苏科技大学学报(自然科学版) 19 6]
[4] Long Z J, Lee S K, Kim J Y 2010 Ocean Engineering 37 418
[5] Marcos S H C, Lopes S R, Viana R L 2003 Chaos, Solitons and Fractals 15 417
[6] Lewis C P, Ucar A, Bishop S R 1998 Transactions of the Institute of Measurement and Control 20 29
[7] Zhang Q, Wang B H, Yang C W 2005 Power System Technology 29 63 (in Chinese) [张强, 王宝华, 杨成梧 2005 电网技术 29 63]
[8] Fadi M A, Mohammad I Y, Hassen M O 2009 Smart Material Structure 19 045013
[9] Lenci S, Rega G 2006 J. Micromechanics and Microengineering 16 390
[10] Fadi M A, Mohammad I Y 2010 Smart Material Structure 19 035016
[11] Shang H L, Xu J 2009 Chaos, Solitons and Fractals 41 1880
[12] Shang H L 2011 Chin. Phys. Lett. 28 010502
[13] Shang H L 2011 Acta Phys. Sin. 60 070501 (in Chinese) [尚慧琳 2011 60 070501]
[14] Sun Z K, Xu W, Yang X L, Fang T 2006 Chaos, Solitons and Fractals 27 705
[15] Gan C B 2006 Nonlin. Dyn. 45 305
[16] Gan C B, He S M 2007 Acta Mech. Sin. 23 577
[17] Stephen W 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos (2nd Ed.) (New York: Springer-Verlag) p62
[18] Hu H Y 2000 Applied Nonlinear Dynamics (Beijing: Aviation Industry Press) p176 (in Chinese) [胡海岩 2000 应用非线性动力学 (北京: 航空工业出版社) 第176页]
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[1] Thompson J M T, Rainey F C T, Soliman M S 1995 Philosophical Transactions of the Royal Society 332 149
[2] Soliman M S 1995 J. Sound Vib. 182 618
[3] Gu J Y, Miao Z H 2005 Journal of Jiangsu University of Science and Technology (Natural Science Edition) 19 6 (in Chinese) [谷家扬, 缪振华 2005 江苏科技大学学报(自然科学版) 19 6]
[4] Long Z J, Lee S K, Kim J Y 2010 Ocean Engineering 37 418
[5] Marcos S H C, Lopes S R, Viana R L 2003 Chaos, Solitons and Fractals 15 417
[6] Lewis C P, Ucar A, Bishop S R 1998 Transactions of the Institute of Measurement and Control 20 29
[7] Zhang Q, Wang B H, Yang C W 2005 Power System Technology 29 63 (in Chinese) [张强, 王宝华, 杨成梧 2005 电网技术 29 63]
[8] Fadi M A, Mohammad I Y, Hassen M O 2009 Smart Material Structure 19 045013
[9] Lenci S, Rega G 2006 J. Micromechanics and Microengineering 16 390
[10] Fadi M A, Mohammad I Y 2010 Smart Material Structure 19 035016
[11] Shang H L, Xu J 2009 Chaos, Solitons and Fractals 41 1880
[12] Shang H L 2011 Chin. Phys. Lett. 28 010502
[13] Shang H L 2011 Acta Phys. Sin. 60 070501 (in Chinese) [尚慧琳 2011 60 070501]
[14] Sun Z K, Xu W, Yang X L, Fang T 2006 Chaos, Solitons and Fractals 27 705
[15] Gan C B 2006 Nonlin. Dyn. 45 305
[16] Gan C B, He S M 2007 Acta Mech. Sin. 23 577
[17] Stephen W 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos (2nd Ed.) (New York: Springer-Verlag) p62
[18] Hu H Y 2000 Applied Nonlinear Dynamics (Beijing: Aviation Industry Press) p176 (in Chinese) [胡海岩 2000 应用非线性动力学 (北京: 航空工业出版社) 第176页]
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