Chaotic threshold for a class of impulsive differential system

详细信息    查看全文

• 作者：RuiLan Tian ; YuFeng Zhou ; BaoLing Zhang ; XinWei Yang
• 关键词：Impulsive differential system ; Non ; smooth homoclinic orbit ; Chaos ; Melnikov method
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：83
• 期：4
• 页码：2229-2240
• 全文大小：1,982 KB
• 参考文献：1.Yang, S.P., Li, S.H., Lu, Y.J.: Dynamics of vehicle-pavement coupled system based on a revised flexible roller contact tire model. Sci. China Ser. E 52, 721–730 (2009)CrossRef
  2.Yang, S.P., Li, S.H., Lu, Y.J.: Investigation on dynamical interaction between a heavy vehicle and road pavement. Veh. Syst. Dyn. 48, 923–944 (2010)CrossRef
  3.Melnikov method for discontinuous planar systems: Kukuc̆ka, P. Nonlinear Anal. Theory Methods Appl. 66, 2698–2719 (2007)CrossRef
  4.Cao, Q.J., Wiercigroch, M., Pavlovskaia, E.: Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics. Philos. Trans. R. Soc. A 366, 635–652 (2008)CrossRef
  5.Kunze, M.: Non-smooth Dynamical Systems. Springer, Berlin (2000)CrossRef
  6.Battelli, F., Feckan, M.: Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems. Phys. D 241, 1962–1975 (2012)CrossRef
  7.Shi, L.S., Zou, Y.K., KÄupper, T.: Melnikov method and detection of chaos for non-smooth systems. Acta Math. Appl. Sin. E 29, 881–896 (2013)
  8.Li, S.B., Zhang, W., Hao, Y.X.: Melnikov-type method for a class of discontinuous planar systems and applications. Int. J. Bifurc. Chaos 24, 14500221–10 (2014)
  9.Li, S.B., Shen, C., Zhang, W., Hao, Y.X.: Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping. Nonlinear Dyn. 79, 2395–2406 (2015)CrossRef
  10.Du, Z.D., Zhang, W.N.: Melnikov method for homoclinic bifurcation in nonlinear impact oscillators. Comput. Math. Appl. 50, 445–458 (2005)CrossRef
  11.Du, Z., Li, Y.R., Zhang, W.N.: Type I periodic motions for nonlinear impact oscillators. Nonlinear Anal. Theory Methods Appl. 67, 1344–1358 (2007)CrossRef
  12.Li, Y.R., Du, Z.D., Zhang, W.N.: Asymmetric type II periodic motions for nonlinear impact oscillators. Nonlinear Anal. Theory Methods Appl. 68, 2681–2696 (2008)CrossRef
  13.Xu, W., Feng, J.Q., Rong, H.W.: Melnikov’s method for a general nonlinear vibro-impact oscillator. Nonlinear Anal. Theory Methods Appl. 71, 418–426 (2009)CrossRef
  14.Du, Z.D., Li, Y.R., Shen, J., Zhang, W.N.: Impact oscillators with homoclinic orbit tangent to the wall. Phys. D 245, 19–33 (2013)CrossRef
  15.Carmona, V., Fernandez-Garcia, S., Freire, E., Torres, F.: Melnikov theory for a class of planar hybrid systems. Phys. D 248, 44–54 (2013)CrossRef
  16.Wojewoda, B.J., Stefan̆ski, A., Wiercigroch, M., Kapitaniak, T.: Hysteretic effects of dry friction: modeling and experimental studies. Philos. Trans. R. Soc. A 366, 747–765 (2008)
  17.Chávez, J.P., Pavlovskaia, E., Wiercigroch, M.: Bifurcation analysis of a piecewise-linear impact oscillator with drift. Nonlinear Dyn. 77, 229–230 (2014)CrossRef
  18.Cao, Q.J., Han, Y.W., Liang, T.W., Wiercigroch, M., Piskarev, S.: Multiple buckling and codimension-three bifurcation phenomena of a nonlinear oscillator. Int. J. Bifur. Chaos 24, 14300051–17 (2014)CrossRef
  19.Tian, R.L., Wu, Q.L., Yang, X.W., Si, C.D.: Chaotic threshold for the smooth-and-discontinuous oscillator under constant excitations. Eur. Phys. J. Plus 128, 801–812 (2013)CrossRef
  20.Witelski, T., Virgin, L.N., George, C.: A driven system of impacting pendulums: experiments and simulations. J. Sound Vib. 333, 1734–1753 (2014)CrossRef
  21.Yu, X.L., Wang, J.R.: Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces. Commun. Nonlinear Sci. 22, 980–989 (2015)CrossRef
  22.Wu, R.H., Zou, X.L., Wang, K.: Asymptotic behavior of a stochastic non-autonomous predator-prey model with impulsive perturbations. Commun. Nonlinear Sci. 20, 965–974 (2015)CrossRef
  23.Chen, H.W., Li, J.L., He, Z.M.: The existence of subharmonic solutions with prescribed minimal period for forced pendulum equations with impulses. Appl. Mathods Model. 37, 4189–4198 (2013)CrossRef
  24.Ruan, J., Lin, W.: Chaos in a class of impulsive differential equation. Commun. Nonlinear Sci. 4, 165–169 (1999)CrossRef
  25.Yang, Q.G., Jiang, G.R., Zhou, T.S.: Complex dynamics of a Hamiltonian system under impulsive control. Int. J. Bifur. Chaos 22, 12502971–12 (2012)
  26.Battelli, F., Feckan, M.: Chaos in singular impulsive ODE. Nonlinear Anal. Theory Methods Appl. 28, 655–671 (1997)
  27.Wang, W., Zhang, Q.C., Feng, J.J.: Global bifurcations of strongly nonlinear oscillator induced by parametric and external excitation. Sci. China Technol. Sci. 54, 1986–1991 (2011)CrossRef
  28.Feng, J.J., Zhang, Q.C., Wang, W.: Chaos of several typical asymmetric systems. Chaos Solitons Fractals 45, 950–958 (2012)CrossRef
  29.Han, J.X., Zhang, Q.C., Wang, W.: Design considerations on large amplitude vibration of a doubly clamped microresonator with two symmetrically located electrodes. Commun. Nonlinear Sci. Numer. Simul. 22, 492–510 (2015)CrossRef
  
• 作者单位：RuiLan Tian (1)
  YuFeng Zhou (1)
  BaoLing Zhang (1)
  XinWei Yang (2)
  
  1. Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China
  2. School of Traffic, Shijiazhuang Institute of Railway Technology, Shijiazhuang, 050041, China
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

A kind of impulsive differential system is constructed by the use of the non-smooth pendulum which is composed of a rigid wall and a pendulum. The pendulum is subjected to different types of impulsive excitations, which lead to the non-smooth homoclinic orbits. Specifically, the existence of non-smooth homoclinic orbits depends on both the classical heteroclinic orbits and type II periodic orbits. When the pendulum moves to the highest point, an impact impulsive excitation is considered. While the orbits arrived at the lowest point, other types of impulsive excitations are introduced. Hence, these non-smooth homoclinic orbits hold two classes of jump discontinuities. One is related to the direction of velocity and the other administered by the magnitude of velocity. In order to illustrate the criteria for chaotic motion of this kind of system, the well-known Melnikov theory for the smooth system is extended applying the Hamiltonian function, which reveals the effects of these impulsive excitations on the behaviors of nonlinear dynamical systems. The efficiency of the criteria for bifurcation and chaos mentioned above is verified by the phase portrait, Poincaré surface of section, and bifurcation diagrams. Keywords Impulsive differential system Non-smooth homoclinic orbit Chaos Melnikov method