Chin. Phys. Lett.  2011, Vol. 28 Issue (1): 010502    DOI: 10.1088/0256-307X/28/1/010502
GENERAL |
Control of Fractal Erosion of Safe Basins in a Holmes–Duffing System via Delayed Position Feedback
SHANG Hui-Lin
School of Mechanical and Automation Engineering, Shanghai Institute of Technology, Shanghai 200235
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SHANG Hui-Lin 2011 Chin. Phys. Lett. 28 010502
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Abstract A linear delayed position feedback control is applied to control the erosion of safe basins in a Holmes–Duffing system. The conditions of fractal erosion of the safe basin of the controlled system on the basis that the range of time delay leading to good control is obtained by the Melnikov method. It is found that the increasing time delay can reduce the basin erosion under a weak and positive feedback gain. Then the evolutions of safe basins with time delay are presented in detail by the fourth Runge-Kutta and Monte-Carlo methods, which shows that the safe basin of the controlled Holmes–Duffing system can be expanded, and its fractal can be reduced by the increasing time delay. These results suggest that delayed position feedbacks can be used as a good approach to control the erosion of safe basins.
Keywords: 05.45.-a      05.45.Gg      05.45.Df      02.30.Ks     
Received: 17 August 2010      Published: 23 December 2010
PACS:  05.45.-a (Nonlinear dynamics and chaos)  
  05.45.Gg (Control of chaos, applications of chaos)  
  05.45.Df (Fractals)  
  02.30.Ks (Delay and functional equations)  
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https://cpl.iphy.ac.cn/10.1088/0256-307X/28/1/010502       OR      https://cpl.iphy.ac.cn/Y2011/V28/I1/010502
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SHANG Hui-Lin
[1] Freitas M, Viana R and Grebogi C 2003 Chaos, Solitons & Fractals 18 829
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[4] Rega G and Lenci S 2005 Nonlinear Analysis 63 902
[5] Xu J, Lu Q S and Huang K L 1996 Acta Mech. Sin. 12 281
[6] Lenci S and Rega G 2003 J. Vibration and Control 9 281
[7] Gong P L, Xu J X and Sun Z Z 2001 Acta Phys. Sin. 50 841 (in Chinese)
[8] Rong H W, Wang X D and Xu W 2008 Acta Phys. Sin. 57 1506 (in Chinese)
[9] Gan C B 2005 Chaos, Solitons & Fractals 25 1069
[10] Gan C B and Guo T Y 2007 J. Vibration and Shock 26 112 (in Chinese)
[11] Sun Z K, Xu W and Yang X L 2006 Chaos, Solitons & Fractals 27 705
[12] Pyragas K 1995 Phys. Lett. A 206 323
[13] Shang H L and Xu J 2009 Chaos, Solitons & Fractal 41 1880
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