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Anti-control of bifurcation, as an inverse problem of conventional bifurcation analysis, is aimed at creating a certain bifurcation with desired dynamic properties at a pre-specified system parameter location via control. The main purpose of this paper is to address the problem of anti-control of Neimark-Sacker bifurcation of a three-degree-of-freedom vibro-impact system with clearance (i.e., the second Hopf bifurcation of the original system), which may be viewed as a design approach to creating a quasi-periodic impact motion (or torus solution) at a specified system parameter location via control. Firstly, in the premise of no change of periodic solutions of the original system, when the difficulties that are brought about by the implicit Poincaré map of the vibro-impact system are considered, a linear feedback controller is added to the original system and a six-dimensional Poincaré map of the close-loop control system is established. In order to design a desired bifurcation solution by control, the multiple control gains are used to tune the existence of this bifurcation based on the corresponding critical criterion. However, for six-dimensional map of the vibro-impact system in the paper, the analytical expressions of all eigenvalues of Jacobi matrix with respect to parameters are unavailable. This implies that when the classical critical criterion described by the properties of eigenvalues is used, we have to numerically compute eigenvalues point by point and check their properties to search for the control gains. So, the numerical calculation is a laborious job in the process of determining the control gains. To overcome the difficulty originating from the classical bifurcation criterion, the explicit critical criterion without using eigenvalue calculation of high-dimensional map is used to obtain the controlling parameters area when quasi-periodic impact motion occurs. Then, the stability of quasi-periodic bifurcation solution is analyzed by utilizing the center manifold and normal formal theory. Finally the numerical experiments verify that the stable quasi-periodic impact motion can be generated at a designated system parameter point by the proposed control.
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Keywords:
- Neimark-Sacker bifurcation /
- anti-controlling bifurcation /
- stability /
- vibro-impact system
[1] Shaw S W, Rand R H 1989 Int. J. Non-linear Mech. 24 41
[2] Hang W H, Chen B, Wang Z L 1994 Recent Developments of General Mechanics (Dynamic, Vibration and Control) (Beijing: Science Press) pp198-204 (in Chinese)[黄文虎, 陈滨, 王照林1994 一般力学(动力学、振动 与控制) 最新进展(北京: 科学出版社) 第198—204 页]
[3] Luo G W, Lv X H, Zhu X F 2008 Int. J. Mech. Sci. 50 214
[4] Jin L, Lu Q S, Wang Q 2005 Appl. Math. Mech. 26 810 (in Chinese) [金俐, 陆启韶, 王琪2005 应用数学和力学26 810]
[5] Zhang H, Ding W C, Li F 2011 Eng. Mech. 28 209 (in Chinese) [张惠, 丁旺才, 李飞2011 工程力学28 209]
[6] Yue Y, Xie J H 2008 J. Sound Vib. 314 228
[7] Li Q H, Tan J Y 2011 Chin. Phys. B 20 040505
[8] Feng J Q, Liu J L 2015 Chaos Soliton. Fract. 73 10
[9] Chai L, Wu X M 2014 J. Xiamen Univ. (Natural Science) 53 508 (in Chinese) [柴林, 吴晓明2014 厦门大学学 报(自然科学版) 53 508]
[10] Chen D S, Wang H O, Chen G R 2001 IEEE Trans. Circ. Syst. I 48 661
[11] Chen Z, Yu P 2005 Chaos Soliton. Fract. 26 1231
[12] Yu S M, Chen G R 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 2617
[13] Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时2008 57 6162]
[14] Zhang L M, Zhang J W, Wu R H 2014 Acta Phys. Sin. 63 160505 (in Chinese) [张玲梅, 张建文, 吴润衡2014 物 理学报63 160505]
[15] Cheng Z S 2010 Neurocomputing 73 3139
[16] Wu Z Q, Sun L M 2011 Acta Phys. Sin. 60 050504 (in Chinese) [吴志强, 孙立明2011 60 050504]
[17] Wen G L 2005 Phys. Rev. E 72 026201
[18] Luo G W, Xie J H 2004 Periodic Motion and Bifurcation of Vibro-impact System (Beijing: Science Press) pp118-119 (in Chinese) [罗冠炜, 谢建华2004 碰撞振动系 统的周期运动和分岔(北京: 科学出版社) 第118—119 页]
[19] D'Amico M B, Moiola J L, Paolini E E 2003 Dynam. Conti. Dis. Ser. B 10 781
[20] Kuznetsov Y A 1998 Elements of Applied Bifurcation Theory (2nd Ed.) (New York: Springer-Verlag) pp185-187
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[1] Shaw S W, Rand R H 1989 Int. J. Non-linear Mech. 24 41
[2] Hang W H, Chen B, Wang Z L 1994 Recent Developments of General Mechanics (Dynamic, Vibration and Control) (Beijing: Science Press) pp198-204 (in Chinese)[黄文虎, 陈滨, 王照林1994 一般力学(动力学、振动 与控制) 最新进展(北京: 科学出版社) 第198—204 页]
[3] Luo G W, Lv X H, Zhu X F 2008 Int. J. Mech. Sci. 50 214
[4] Jin L, Lu Q S, Wang Q 2005 Appl. Math. Mech. 26 810 (in Chinese) [金俐, 陆启韶, 王琪2005 应用数学和力学26 810]
[5] Zhang H, Ding W C, Li F 2011 Eng. Mech. 28 209 (in Chinese) [张惠, 丁旺才, 李飞2011 工程力学28 209]
[6] Yue Y, Xie J H 2008 J. Sound Vib. 314 228
[7] Li Q H, Tan J Y 2011 Chin. Phys. B 20 040505
[8] Feng J Q, Liu J L 2015 Chaos Soliton. Fract. 73 10
[9] Chai L, Wu X M 2014 J. Xiamen Univ. (Natural Science) 53 508 (in Chinese) [柴林, 吴晓明2014 厦门大学学 报(自然科学版) 53 508]
[10] Chen D S, Wang H O, Chen G R 2001 IEEE Trans. Circ. Syst. I 48 661
[11] Chen Z, Yu P 2005 Chaos Soliton. Fract. 26 1231
[12] Yu S M, Chen G R 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 2617
[13] Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时2008 57 6162]
[14] Zhang L M, Zhang J W, Wu R H 2014 Acta Phys. Sin. 63 160505 (in Chinese) [张玲梅, 张建文, 吴润衡2014 物 理学报63 160505]
[15] Cheng Z S 2010 Neurocomputing 73 3139
[16] Wu Z Q, Sun L M 2011 Acta Phys. Sin. 60 050504 (in Chinese) [吴志强, 孙立明2011 60 050504]
[17] Wen G L 2005 Phys. Rev. E 72 026201
[18] Luo G W, Xie J H 2004 Periodic Motion and Bifurcation of Vibro-impact System (Beijing: Science Press) pp118-119 (in Chinese) [罗冠炜, 谢建华2004 碰撞振动系 统的周期运动和分岔(北京: 科学出版社) 第118—119 页]
[19] D'Amico M B, Moiola J L, Paolini E E 2003 Dynam. Conti. Dis. Ser. B 10 781
[20] Kuznetsov Y A 1998 Elements of Applied Bifurcation Theory (2nd Ed.) (New York: Springer-Verlag) pp185-187
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