一般脉冲系统的Melnikov函数构造方法及其应用
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摘要
利用摄动法给出了一般脉冲系统Melnikov函数构造方法,得到脉冲信号作用下一般非线性系统Melnikov方法.为考察方法的有效性,将方法应用到脉冲信号作用下Duffing系统的混沌预测中去,通过方法得到脉冲信号作用下Duffing系统出现混沌的阈值曲线,数值实验结果验证理论结果的正确性.
The Construction method of general impulsive system is discussed in this paper using perturbation method,and an analytical method is given for the studying of impulsive system,and the necessary condition of chaos appearance in impulsive system is derived.Duffing system with impulsive signals is employed to show the efficiency of this method in the end.
The Construction method of general impulsive system is discussed in this paper using perturbation method,and an analytical method is given for the studying of impulsive system,and the necessary condition of chaos appearance in impulsive system is derived.Duffing system with impulsive signals is employed to show the efficiency of this method in the end.
引文
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[11]Xu W,Feng J Q,and Rong H W.Melnikov method for a general nonlinear Vibro-impact oscillator[J].Nonlinear Analysis,2009,71(9):41 8-426.
[12]Tian R L,Zhou Y F,Wang Q B,and Zhang L L.Bifurcation and chaotic threshold of Duffing system with jump discontinuities[J].The European Physical Journal Plus,2016,131(1):15.
[13]Tian R L,Zhou Y F,Zhang B L,and Yang X W.Chaotic threshold for a kind of impulsive differential system[J].Nonlinear Dynamics,2016,83(4):2229-2240.
[14]Tian R L,Zhou Y F,Wang Y,Feng W,and Yang X.Chaotic threshold for non-smooth systems with multiple impulse effect[J].Nonlinear Dynamics,2016,85(3):1849-1863.
[15]Tian R L,Wu Q,Yang X,and Si C.Chaotic threshold for the smooth-and-discontinuous oscillator under constant excitations[J].The European Physical Journal Plus,2013,128(7):80.
[16]刘曾荣.混沌研究中的解析方法[M].上海:上海大学出版社,2002.
[2]Sun Z K,Xiao Y Z,and Xu W.Delay-induced stochastic bifurcations in a bistable system under white noise[J].Chaos,2015,25(8):083102.
[3]Niu Y J,Liao D,and Wang P.Stochastic asymptotical stability for stochastic impulsive differential equations and it is application to chaos synchronization[J].Communication in Nonlinear and Numerical Simulation,2012,17(2):505-512.
[4]Ding J,Cao J,Feng G.Alsaedi A.Stability analysis of delayed impulsive systems and applications[J].Circuits Systems and Signal processing,2018,37(3):1062-1080.
[5]Feckan M,Wang J,and Zhou Y.Periodic solutions for nonlinear evolution equations with noninstantaneous impulses[J].Nonautonomous Dynamical Systems,2014,83(1):93-101.
[6]Feckan M,Zhou Y,and Wang J.On the concept and existence of solution for impulsive fractional differential equations[J].Communication in Nonlinear Science and Numerical Simulation,2012,17(7):3050-3060.
[7]Wang J,Feckan M,and Zhou Y.A survey on impulsive fractional differential equations[J].Fractional Calculus and Applied Analysis,2016,19(4):806-831.
[8]Feckan M,Melnikov functions for singularly perturbed ordinary differential equations[J].NonlinearAn alysis Theory Method and Applications,1992,19(4):393-401.
[9]Kukueka P.Melnikov method for discontinuous planar systems[J].Nonlinear Analysis,2007,66(12):2698-2719.
[10]Battelli F,Feckan M.Nonlinear homoclinic orbits,Melnikov functions and chaos in discontinuous systems[J].Physica D,2012,241(22):1962-1975.
[11]Xu W,Feng J Q,and Rong H W.Melnikov method for a general nonlinear Vibro-impact oscillator[J].Nonlinear Analysis,2009,71(9):41 8-426.
[12]Tian R L,Zhou Y F,Wang Q B,and Zhang L L.Bifurcation and chaotic threshold of Duffing system with jump discontinuities[J].The European Physical Journal Plus,2016,131(1):15.
[13]Tian R L,Zhou Y F,Zhang B L,and Yang X W.Chaotic threshold for a kind of impulsive differential system[J].Nonlinear Dynamics,2016,83(4):2229-2240.
[14]Tian R L,Zhou Y F,Wang Y,Feng W,and Yang X.Chaotic threshold for non-smooth systems with multiple impulse effect[J].Nonlinear Dynamics,2016,85(3):1849-1863.
[15]Tian R L,Wu Q,Yang X,and Si C.Chaotic threshold for the smooth-and-discontinuous oscillator under constant excitations[J].The European Physical Journal Plus,2013,128(7):80.
[16]刘曾荣.混沌研究中的解析方法[M].上海:上海大学出版社,2002.