Erosion of safe basins in a softening duffing system and its control by elayed position feedback

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Journal of Vibration and Shock ›› 2012, Vol. 31 ›› Issue (8) : 11-15,2.

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Erosion of safe basins in a softening duffing system and its control by elayed position feedback

Shang Hui-lin1, wen yong-peng2

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Abstract

Linear delayed position feedback is induced to a softening Duffing system, and its effect on controlling the erosion of safe basins is investigated. Firstly, the condition of fractal erosion of the basin boundary is obtained. The results show that the increasing of time delay can enhance the threshold of the excitation amplitude of the fractal erosion and reduce the erosion under a weak feedback control. Then the definition of safe basin in delayed feedback controlled system is given according to the idea of projecting infinite initial space of delayed system into finite-dimensional Euclidean Space. Finally, considering time delay as a variable parameter and employing the 4th Runge-Kutta and the point mapping methods, evolutions of safe basin with time delay are presented. It is found that the increasing of time delay within a certain range can reduce the erosion of safe basins effectively. The agreement of the theoretical analysis and numerical simulations verifies the validity of the analytical prediction. All the results have an implication that delayed position feedback can be used as a good approach to control the erosion of safe basins.

Key words

safe basin / Softening Duffing system / fractal / time delay / feedback

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Shang Hui-lin;wen yong-peng. Erosion of safe basins in a softening duffing system and its control by elayed position feedback[J]. Journal of Vibration and Shock, 2012, 31(8): 11-15,2