Anti-controlling Pitchfork bifurcation on Poincar&eacute; map of a three-degree-of-freedom vibro-impact system

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Journal of Vibration and Shock ›› 2016, Vol. 35 ›› Issue (20) : 24-29.

Anti-controlling Pitchfork bifurcation on Poincaré map of a three-degree-of-freedom vibro-impact system

WU Xin1,2 XU Hui-dong3 WEN Gui-lin 2 WEI Ke-xiang 1

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Abstract

In the premise of no change of periodic solutions of the original system and with consideration of the difficulties that given by the implicit Poincaré map of the vibro-impact system, anti-control of Pitchfork bifurcation on Poincaré map of a three-degree-of-freedom vibro-impact system is studied by using linear feedback control method. Firstly, the six-dimensional Poincaré map of close-loop system is established, to overcome the difficulty that the numerical computing method can be only used to determine control gains on basis of the classical critical criteria of Pitchfork bifurcation described by the properties of eigenvalues in six-dimensional map, an explicit Pitchfork critical criterion without using eigenvalues is used to obtain the controlling parameters area of two parameters. Then, the stability of the Pitchfork bifurcation is further analyzed by utilizing the center manifold and normal formal theory. Finally, the numerical experiments verify that the stable Pitchfork bifurcation solutions can be generated at an arbitrary specified parameters point by controlling.

Key words

Pitchfork bifurcation / anti-controlling bifurcation / stability / vibro-impact system

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WU Xin1,2 XU Hui-dong3 WEN Gui-lin 2 WEI Ke-xiang 1. Anti-controlling Pitchfork bifurcation on Poincaré map of a three-degree-of-freedom vibro-impact system[J]. Journal of Vibration and Shock, 2016, 35(20): 24-29

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