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Bursting oscillations and mechanism of sliding movement in piecewise Filippov system

Zhang Zheng-Di Liu Ya-Nan Li Jing Bi Qin-Sheng

Bursting oscillations and mechanism of sliding movement in piecewise Filippov system

Zhang Zheng-Di, Liu Ya-Nan, Li Jing, Bi Qin-Sheng
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  • Since the wide applications in science and engineering, the dynamics of non-smooth system has become one of the key research subjects. Furthermore, the interaction between different scales may result in special movement which can be usually described by the combination of large-amplitude oscillation and small-amplitude one. The influence of multiple scale on the dynamics of non-smooth system has received much attention recently. In this work, we try to explore the bursting oscillations and the mechanism of non-smooth Filippov system coupled by different scales in the frequency domain. Taking the typical periodically excited Duffing's oscillator for example a Filippov system coupled by two scales in the frequency domain is established when the difference in order between the excited frequency and the system natural frequency is obtained by introducing the piecewise control into the state variable and choosing suitable parameters. For the case in which the exciting frequency is far less than the natural frequency, the whole exciting term can be considered as a slow-varying parameter, also called slow subsystem, which leads to a generalized autonomous system, i.e., the fast subsystem. The equilibrium branches and the bifurcations of the fast subsystem along with the variation of the slow-varying parameter in different regions divided according to non-smooth boundary, can be derived. Two typical cases are taken into consideration, in which different distributions of the equilibrium branches and the relevant bifurcations of the fast subsystem may exist. It is pointed out that the variations of the parameters may influence not only the properties of the equilibrium branches, but also the structures of the bursting attractors. Furthermore, since the governing equation alternates between two subsystems located in different regions when the trajectory passes across the non-smooth boundary, the sliding movement along the non-smooth boundary of the trajectory can be observed under the condition of certain parameters. By employing the transformed phase portrait which describes the relationship between the state variable and the slow-varying parameter, the mechanisms of different bursting oscillations and sliding movements are investigated. The results show that bursting oscillations may exist in a non-smooth Filippov system coupled by two scales in the frequency domain. The alternations of the governing equation between different subsystems located in the two neighboring regions along the non-smooth boundary may result in a sliding movement of the trajectory along the non-smooth boundary.
      Corresponding author: Bi Qin-Sheng, qbi@ujs.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11472116, 11472115) and the Key Program of the National Natural Science Foundation of China (Grant No. 11632008).
    [1]

    Duan C, Singh R 2005 J. Sound Vib. 285 1223

    [2]

    Siefert A, Henkel F O 2014 Nucl. Eng. Des. 269 130

    [3]

    Chen Z Y, Wang Y M 2016 J. Henan Sci. Univ. 37 87(in Chinese) [陈章耀, 王亚茗 2016 河南科技大学学报 37 87]

    [4]

    Galvanetto U 2001 J. Sound Vib. 248 653

    [5]

    Carmona V, Fernndez-Garca S, Freire E 2012 Physica D 241 623

    [6]

    Dercole F, Gragnani A, Rinaldi S 2007 Theor. Popul. Biol. 72 197

    [7]

    Zhang S J, Zhou L B, Lu Q S 2007 J. Mech. 39 132(in Chinese) [张思进, 周利彪, 陆启韶 2007 力学学报 39 132]

    [8]

    Zhang X F, Chen X K, Bi Q S 2012 J. Mech. 44 576(in Chinese) [张晓芳, 陈小可, 毕勤胜 2012 力学学报 44 576]

    [9]

    Zhou Z, Tan Y, Xie Y 2016 Mech. Syst. Sig. Process. 83 439

    [10]

    Kahan S, Sicardi-Schifino A C 1999 Physica A 262 144

    [11]

    Baptista M S 1999 Physica D 132 325

    [12]

    Leine R I 2006 Physica D 223 121

    [13]

    Leine R I, Glocker C 2003 Eur. J. Mech. A: Solids 22 193

    [14]

    Leine R I, Campen D H V 2006 Eur. J. Mech. 25 595

    [15]

    Izhikevich E M, Desai N S, Walcott E C 2003 Trends Neurosci. 26 161

    [16]

    Vanag V K, Epstein I R 2011 Phys. Rev. E 84 066209

    [17]

    Jia Z, Leimkuhler B 2003 Future Gener. Comp. Syst. 19 415

    [18]

    Yu B S, Jin D P, Pang Z J 2014 Science China E 8 858(in Chinese) [余本嵩, 金栋平, 庞兆君 2014 中国科学 8 858]

    [19]

    Yang S C, Hong H P 2016 Eng. Struct. 123 490

    [20]

    Ji Y, Bi Q S 2010 Phys. Lett. A 374 1434

    [21]

    Cardin P T, de Moraes J R, da Silva P R 2015 J. Math. Anal. Appl. 423 1166

    [22]

    Hodgkin A L, Huxley A F 1990 Bull. Math. Biol. 52 25

    [23]

    Ferrari F A S, Viana R L, Lopes S R, Stoop R 2015 Neural Networks 66 107

    [24]

    Huang X G, Xu J X, He D H, Xia J L, L Z J 1999 Acta Phys. Sin. 48 1810(in Chinese) [黄显高, 徐健学, 何岱海, 夏军利, 吕泽均 1999 物理学报 48 1810]

    [25]

    Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171

    [26]

    Shimizu K, Saito Y, Sekikawa M 2015 Physica D 241 1518

    [27]

    Han X J, Bi Q S 2012 Int. J. Non Linear Mech. 89 69

    [28]

    Wu T Y, Chen X K, Zhang Z D, Zhang X F, Bi Q S 2017 Acta Phys. Sin. 66 35(in Chinese) [吴天一, 陈小可, 张正娣, 张晓芳, 毕勤胜 2017 物理学报 66 35]

    [29]

    Tana X, Qinc W, Liud X, Jin Y, Jiangb S 2016 J. Nonlinear Sci. Appl. 9 3948

    [30]

    Premraj D, Suresh K, Palanivel J 2017 Commun. Nonlinear Sci. 50 103

    [31]

    Yang X F, Zhang Z D, Li S L 2017 J. Henan Sci. Univ. 38 65(in Chinese) [杨秀芳, 张正娣, 李绍龙 2017 河南科技大学学报 38 65]

    [32]

    Yang X F 2017 M. S. Dissertation (Zhenjiang: Jiangsu University) (in Chinese) [杨秀芳 2017 硕士学位论文(镇江: 江苏大学)]

  • [1]

    Duan C, Singh R 2005 J. Sound Vib. 285 1223

    [2]

    Siefert A, Henkel F O 2014 Nucl. Eng. Des. 269 130

    [3]

    Chen Z Y, Wang Y M 2016 J. Henan Sci. Univ. 37 87(in Chinese) [陈章耀, 王亚茗 2016 河南科技大学学报 37 87]

    [4]

    Galvanetto U 2001 J. Sound Vib. 248 653

    [5]

    Carmona V, Fernndez-Garca S, Freire E 2012 Physica D 241 623

    [6]

    Dercole F, Gragnani A, Rinaldi S 2007 Theor. Popul. Biol. 72 197

    [7]

    Zhang S J, Zhou L B, Lu Q S 2007 J. Mech. 39 132(in Chinese) [张思进, 周利彪, 陆启韶 2007 力学学报 39 132]

    [8]

    Zhang X F, Chen X K, Bi Q S 2012 J. Mech. 44 576(in Chinese) [张晓芳, 陈小可, 毕勤胜 2012 力学学报 44 576]

    [9]

    Zhou Z, Tan Y, Xie Y 2016 Mech. Syst. Sig. Process. 83 439

    [10]

    Kahan S, Sicardi-Schifino A C 1999 Physica A 262 144

    [11]

    Baptista M S 1999 Physica D 132 325

    [12]

    Leine R I 2006 Physica D 223 121

    [13]

    Leine R I, Glocker C 2003 Eur. J. Mech. A: Solids 22 193

    [14]

    Leine R I, Campen D H V 2006 Eur. J. Mech. 25 595

    [15]

    Izhikevich E M, Desai N S, Walcott E C 2003 Trends Neurosci. 26 161

    [16]

    Vanag V K, Epstein I R 2011 Phys. Rev. E 84 066209

    [17]

    Jia Z, Leimkuhler B 2003 Future Gener. Comp. Syst. 19 415

    [18]

    Yu B S, Jin D P, Pang Z J 2014 Science China E 8 858(in Chinese) [余本嵩, 金栋平, 庞兆君 2014 中国科学 8 858]

    [19]

    Yang S C, Hong H P 2016 Eng. Struct. 123 490

    [20]

    Ji Y, Bi Q S 2010 Phys. Lett. A 374 1434

    [21]

    Cardin P T, de Moraes J R, da Silva P R 2015 J. Math. Anal. Appl. 423 1166

    [22]

    Hodgkin A L, Huxley A F 1990 Bull. Math. Biol. 52 25

    [23]

    Ferrari F A S, Viana R L, Lopes S R, Stoop R 2015 Neural Networks 66 107

    [24]

    Huang X G, Xu J X, He D H, Xia J L, L Z J 1999 Acta Phys. Sin. 48 1810(in Chinese) [黄显高, 徐健学, 何岱海, 夏军利, 吕泽均 1999 物理学报 48 1810]

    [25]

    Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171

    [26]

    Shimizu K, Saito Y, Sekikawa M 2015 Physica D 241 1518

    [27]

    Han X J, Bi Q S 2012 Int. J. Non Linear Mech. 89 69

    [28]

    Wu T Y, Chen X K, Zhang Z D, Zhang X F, Bi Q S 2017 Acta Phys. Sin. 66 35(in Chinese) [吴天一, 陈小可, 张正娣, 张晓芳, 毕勤胜 2017 物理学报 66 35]

    [29]

    Tana X, Qinc W, Liud X, Jin Y, Jiangb S 2016 J. Nonlinear Sci. Appl. 9 3948

    [30]

    Premraj D, Suresh K, Palanivel J 2017 Commun. Nonlinear Sci. 50 103

    [31]

    Yang X F, Zhang Z D, Li S L 2017 J. Henan Sci. Univ. 38 65(in Chinese) [杨秀芳, 张正娣, 李绍龙 2017 河南科技大学学报 38 65]

    [32]

    Yang X F 2017 M. S. Dissertation (Zhenjiang: Jiangsu University) (in Chinese) [杨秀芳 2017 硕士学位论文(镇江: 江苏大学)]

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Publishing process
  • Received Date:  10 November 2017
  • Accepted Date:  13 January 2018
  • Published Online:  05 June 2018

Bursting oscillations and mechanism of sliding movement in piecewise Filippov system

    Corresponding author: Bi Qin-Sheng, qbi@ujs.edu.cn
  • 1. Faculty of Science, Jiangsu University, Zhenjiang 212013, China;
  • 2. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos. 11472116, 11472115) and the Key Program of the National Natural Science Foundation of China (Grant No. 11632008).

Abstract: Since the wide applications in science and engineering, the dynamics of non-smooth system has become one of the key research subjects. Furthermore, the interaction between different scales may result in special movement which can be usually described by the combination of large-amplitude oscillation and small-amplitude one. The influence of multiple scale on the dynamics of non-smooth system has received much attention recently. In this work, we try to explore the bursting oscillations and the mechanism of non-smooth Filippov system coupled by different scales in the frequency domain. Taking the typical periodically excited Duffing's oscillator for example a Filippov system coupled by two scales in the frequency domain is established when the difference in order between the excited frequency and the system natural frequency is obtained by introducing the piecewise control into the state variable and choosing suitable parameters. For the case in which the exciting frequency is far less than the natural frequency, the whole exciting term can be considered as a slow-varying parameter, also called slow subsystem, which leads to a generalized autonomous system, i.e., the fast subsystem. The equilibrium branches and the bifurcations of the fast subsystem along with the variation of the slow-varying parameter in different regions divided according to non-smooth boundary, can be derived. Two typical cases are taken into consideration, in which different distributions of the equilibrium branches and the relevant bifurcations of the fast subsystem may exist. It is pointed out that the variations of the parameters may influence not only the properties of the equilibrium branches, but also the structures of the bursting attractors. Furthermore, since the governing equation alternates between two subsystems located in different regions when the trajectory passes across the non-smooth boundary, the sliding movement along the non-smooth boundary of the trajectory can be observed under the condition of certain parameters. By employing the transformed phase portrait which describes the relationship between the state variable and the slow-varying parameter, the mechanisms of different bursting oscillations and sliding movements are investigated. The results show that bursting oscillations may exist in a non-smooth Filippov system coupled by two scales in the frequency domain. The alternations of the governing equation between different subsystems located in the two neighboring regions along the non-smooth boundary may result in a sliding movement of the trajectory along the non-smooth boundary.

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