新混沌系统及其分数阶系统的自适应同步控制
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摘要
研究一个存在共存吸引子的混沌系统及相应的分数阶系统的自适应同步问题.首先,提出了一个新的具有双翼和四翼吸引子共存的混沌系统,对系统的动力学特性进行了分析,找到了系统的拓扑马蹄和拓扑熵,从而验证了系统具有混沌特性;然后,根据该系统构建了一个亦存在两个孤立的双翼吸引子以及四翼吸引子的分数阶系统.最后,采用分数阶Lyapunov稳定性理论以及自适应控制方法,对分数阶系统的自适应同步问题进行了研究.仿真结果表明,控制参数k越大,系统同步速度越快;控制参数λ越大,系统参数识别的速度越快.
A novel chaotic system with coexisting attractors and the adaptive synchronization control of the corresponding fractional order chaotic system are studied. Firstly, a novel chaotic system with coexisting attractors of double-wing and four-wing is proposed. The dynamic characteristics of the system are analyzed, and the topological horseshoe and topological entropy of the system are found, which verifies that the system is chaotic. Secondly, a fractional order system with two isolated double-wing and four-wing attractors is constructed based on this system. Finally, the adaptive synchronization problem of the fractional order system is investigated by means of fractional order Lyapunov stability theory and adaptive control method. Simulation results show that the larger the control parameter k, the faster the system synchronization speed,and the larger the control parameter λ, the faster the identification of the system parameters.
A novel chaotic system with coexisting attractors and the adaptive synchronization control of the corresponding fractional order chaotic system are studied. Firstly, a novel chaotic system with coexisting attractors of double-wing and four-wing is proposed. The dynamic characteristics of the system are analyzed, and the topological horseshoe and topological entropy of the system are found, which verifies that the system is chaotic. Secondly, a fractional order system with two isolated double-wing and four-wing attractors is constructed based on this system. Finally, the adaptive synchronization problem of the fractional order system is investigated by means of fractional order Lyapunov stability theory and adaptive control method. Simulation results show that the larger the control parameter k, the faster the system synchronization speed,and the larger the control parameter λ, the faster the identification of the system parameters.
引文
[1]LI Zhenbo,TANG Jiashi.Chaotic synchronization with parameter perturbation and its secure communication scheme[J].Control Theory&Applications,2014,31(5):592–600.(李震波,唐驾时.参数扰动下的混沌同步控制及其保密通信方案[J].控制理论与应用,2014,31(5):592–600.)
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[4]LORENZ E N.Deterministic nonperodic flow[J].Journal of the Atmospheric Sciences,1963,20(1):130–141.
[5]CHEN G R,UETA T.Yet another chaotic attractor[J].International Journal of Bifurcation&Chaos,1999,9(7):1465–1466.
[6]L¨U J H,CHEN G R.A new chaotic attractor coined[J].International Journal of Bifurcation&Chaos,2002,12(3):659–661.
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[10]VAIDYANATHAN S.Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters[J].Journal of Engineering Science&Technology Review,2013,6(4):53–65.
[11]VAIDYANATHAN S.A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities[J].Far East Journal of Mathematical Sciences,2014,84(2):219–226.
[12]PEHLIVAN I,MOROZ I M,VAIDYANATHAN S.Analysis,synchronization and circuit design of a novel butterfly attractor[J].Journal of Sound&Vibration,2014,333(20):5077–5096.
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[15]YUAN F,WANG G Y,SHEN Y R,et al.Coexisting attractors in a memcapacitor-based chaotic oscillator[J].Nonlinear Dynamics,2016,86(1):37–50.
[16]LAI Q,CHEN S M.Coexisting attractors generated from a new 4D smooth chaotic system[J].International Journal of Control Automation&Systems,2016,14(4):1124–1131.
[17]KENGNE J,NJITACKE Z T,FOTSIN H B.Dynamical analysis of a simple autonomous jerk system with multiple attractors[J].Nonlinear Dynamics,2016,83(1/2):751–765.
[18]PHAM V T,VOLOS C,JAFARI S,et al.Coexistence of hidden chaotic attractors in a novel no-equilibrium system[J].Nonlinear Dynamics,2017,87(3):2001–2010.
[19]JIA Hongyan,CHEN Zengqiang,XUE Wei.Analysis and circuit implementation for the fractional-order lorenz system[J].Acta Physica Sinica,2013,62(14):140503.(贾红艳,陈增强,薛薇.分数阶Lorenz系统的分析及电路实现[J].物理学报,2013,62(14):140503.)
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[23]BEGHIN L,ORSINGHER E.The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation[J].Fractional Calculus&Applied Analysis,2003,6(2):187–204.
[24]GAFIYCHUK V,DATSKO B.Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems[J].Computers&Mathematics with Applications,2010,59(3):1101–1107.
[25]ZHANG W,CAI X,HOLM S.Time-fractional heat equations and negative absolute temperatures[J].Computers and Mathematics with Applications,2014,67(1):164–171.
[26]ZHOU T S,LI C P.Synchronization in fractional-order differential systems[J].Physica D:Nonlinear Phenomena,2005,212(1/2):111–125.
[27]YAN Xiaomei,LIU Ding,GUO Huijun.Chaos control of fractional order Chen system via radial basis function neural network[J].Control Theory&Applications,2010,27(3):344–349.(阎晓妹,刘丁,郭会军.分数阶Chen混沌系统的径向基函数神经滑模控制[J].控制理论与应用,2010,27(3):344–349.)
[28]SUN Meimei,HU Yun’an,WEI Jianming.Adaptive repetitive learning synchronization of uncertain fractional order multi-scroll chaotic systems[J].Control Theory&Applications,2016,33(7):936–944.(孙美美,胡云安,韦建明.不确定分数阶多涡卷混沌系统自适应重复学习同步控制[J].控制理论与应用,2016,33(7):936–944.)
[29]BHALEKAR S,DAFTARDAR-GEJJI V.Synchronization of different fractional order chaotic systems using active control[J].Communications in Nonlinear Science&Numerical Simulation,2010,15(11):3536–3546.
[30]DENG W H.Generalized synchronization in fractional order systems[J].Physical Review E,2007,75(5):49–56.
[31]ADLOO H,ROOPAEI M.Review article on adaptive synchronization of chaotic systems with unknown parameters[J].Nonlinear Dynamics,2011,65(1/2):141–159.
[32]TAN W,WANG Y N,ZENG Z F,et al.Adaptive regulation of uncertain chaos with dynamic neural networks[J].Chinese Physics,2004,13(4):459–463.
[33]ZHANG Ruoxun,YANG Shiping.Designing synchronization schemes for a fractional-order hyperchaotic system[J].Acta Physica Sinica,2008,57(11):6837–6843.(张若洵,杨世平.一个分数阶新超混沌系统的同步[J].物理学报,2008,57(11):6837–6843.)
[34]JIA L X,DAI H,HUI M.Nonlinear feedback synchronisation control between fractional-order and integer-order chaotic systems[J].Chinese Physics B,2010,19(11):194–199.
[35]FU J,YU M,MA T D.Modified impulsive synchronization of fractional order hyperchaotic chen systems[J].Chinese Physics B,2011,20(12):160–166.
[36]BAO Bocheng.Introduction to Chaotic Circuits[M].Beijing:Science Press,2013:46–49.(包伯成.混沌电路导论[M].北京:科学出版社,2013:46–49.)
[37]TAVAZOEI M S,HAERI M.A necessary condition for double scroll attractor existencein fractional-order systems[J].Physics Letters A,2007,367(1):102–113.
[38]ZHOU Ping,WEI Lljia,CHENG Xuefeng.A fractional-order chaotic system and its simulation result by electronic workbench(EWB)[J].Journal of Southwest University,2009,31(1):69–74.(周平,危丽佳,程雪峰.一个分数阶混沌系统及其EWB仿真结果[J].西南大学学报,2009,31(1):69–74.)
[39]MATIGNON D.Stability results for fractional differential equations with applications to control processing[J].Computational Engineering in Systems Applications,1996,2:963–968.
[40]OGATA K.Modern Control Engineering[M].Englewood Cliffs,New Jersey:Prentice-Hall,1990.
[41]LI C P,PENG G J.Chaos in Chen’s system with a fractional order[J].Chaos Solitons&Fractals,2004,22(2):443–450.
[42]ZHANG W W,ZHOU S B,LIAO X F,et al.Estimate the largest Lyapunov exponent of fractional-order systems[C]//International Conference on Communications,Circuits and Systems.Fujian,China:IEEE,2008:1121–1124.
[43]ZHOU P,HUANG K.A new 4-D non-equilibrium fractional-order chaotic system and its circuit implementation[J].Communications in Nonlinear Science&Numerical Simulation,2014,19(6):2005–2011.
[44]KIM H S,EYKHOLT R,SALAS J D.Nonlinear dynamics,delay times,and embedding windows[J].Physica D:Nonlinear Phenomena,1999,127(1/2):48–60.
[45]SATO S,SANO M,SAWADA Y.Practical methods of measuring the generalized dimension and the largest lyapunov exponent in high dimensional chaotic systems[J].Progress of Theoretical Physics,1987,77(1):1–5.
[46]HU Jianbing,HAN Yan,ZHAO Lingdong.A novel stablility theorem for fractional systems and its applying in synchronizing fractional chaotic system based on back-stepping approach[J].Acta Physica Sinica,2009,58(4):2235–2239.(胡建兵,韩焱,赵灵冬.一种新的分数阶系统稳定理论及在backstepping方法同步分数阶混沌系统中的应用[J].物理学报,2009,58(4):2235–2239.)
[47]MA Tiedong,JIANG Weibo,FU Jie,et al.Adaptive synchronization of a class of fractional-order chaotic systems[J].Acta Physica Sinica,2012,61(16):593–598.(马铁东,江伟波,浮洁,等.一类分数阶混沌系统的自适应同步[J].物理学报,2012,61(16):593–598.)
[2]RIAZ A,ALI M.Chaotic communications,their applications and advantages over traditional methods of communication[C]//International Symposium on Communication Systems,Networks&Digital Signal Processing.Austria:IEEE,2008:21–24.
[3]ZHAO Wenli,FAN Jian,WU Min,et al.Self-tracing-frequency control in weak signal chaotic detection[J].Control Theory&Applications,2014,31(2):250–255.(赵文礼,范剑,吴敏,等.微弱信号混沌检测的自跟踪扫频控制方法[J].控制理论与应用,2014,31(2):250–255.)
[4]LORENZ E N.Deterministic nonperodic flow[J].Journal of the Atmospheric Sciences,1963,20(1):130–141.
[5]CHEN G R,UETA T.Yet another chaotic attractor[J].International Journal of Bifurcation&Chaos,1999,9(7):1465–1466.
[6]L¨U J H,CHEN G R.A new chaotic attractor coined[J].International Journal of Bifurcation&Chaos,2002,12(3):659–661.
[7]L¨U J H,CHEN G R,CHENG D Z,et al.Bridge the gap between the Lorenz system and the Chen system[J].International Journal of Bifurcation and Chaos,2002,12(12):2917–2926.
[8]VAIDYANATHAN S,PEHLIVAN I.Analysis,control,synchronization,and circuit design of a novel chaotic system[J].Mathematical&Computer Modelling,2012,55(7/8):1904–1915.
[9]VAIDYANATHAN S.Analysis and anti-synchronization of a novel chaotic system via active and adaptive controllers[J].Journal of Engineering Science&Technology Review,2013,6(4):45–52.
[10]VAIDYANATHAN S.Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters[J].Journal of Engineering Science&Technology Review,2013,6(4):53–65.
[11]VAIDYANATHAN S.A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities[J].Far East Journal of Mathematical Sciences,2014,84(2):219–226.
[12]PEHLIVAN I,MOROZ I M,VAIDYANATHAN S.Analysis,synchronization and circuit design of a novel butterfly attractor[J].Journal of Sound&Vibration,2014,333(20):5077–5096.
[13]SHIMIZU T,MORIOKA N.On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model[J].Physics Letters A,1980,76(3):201–204.
[14]ARGYRIS J,ANDREADIS I.On the influence of noise on the coexistence of chaotic attractors[J].Chaos Solitons&Fractals,2000,11(6):941–946.
[15]YUAN F,WANG G Y,SHEN Y R,et al.Coexisting attractors in a memcapacitor-based chaotic oscillator[J].Nonlinear Dynamics,2016,86(1):37–50.
[16]LAI Q,CHEN S M.Coexisting attractors generated from a new 4D smooth chaotic system[J].International Journal of Control Automation&Systems,2016,14(4):1124–1131.
[17]KENGNE J,NJITACKE Z T,FOTSIN H B.Dynamical analysis of a simple autonomous jerk system with multiple attractors[J].Nonlinear Dynamics,2016,83(1/2):751–765.
[18]PHAM V T,VOLOS C,JAFARI S,et al.Coexistence of hidden chaotic attractors in a novel no-equilibrium system[J].Nonlinear Dynamics,2017,87(3):2001–2010.
[19]JIA Hongyan,CHEN Zengqiang,XUE Wei.Analysis and circuit implementation for the fractional-order lorenz system[J].Acta Physica Sinica,2013,62(14):140503.(贾红艳,陈增强,薛薇.分数阶Lorenz系统的分析及电路实现[J].物理学报,2013,62(14):140503.)
[20]CHEN Xiangrong,LIU Chongxin,WANG Faqiang,et al.Study on the fractional-order Liu chaotic system with circuit experiment and its control[J].Acta Physica Sinica,2008,57(3):1416–1422.(陈向荣,刘崇新,王发强,等.分数阶Liu混沌系统及其电路实验的研究与控制[J].物理学报,2008,57(3):1416–1422.)
[21]MAINARDI F.The fundamental solutions for the fractional diffusion-wave equation[J].Applied Mathematics Letters,1996,9(6):23–28.
[22]GORENFLO R,MAINARDI F,MORETTI D,et al.Fractional diffusion:probability distributions and random walk models[J].Physica A:Statistical Mechanics&Its Applications,2002,305(1):106–112.
[23]BEGHIN L,ORSINGHER E.The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation[J].Fractional Calculus&Applied Analysis,2003,6(2):187–204.
[24]GAFIYCHUK V,DATSKO B.Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems[J].Computers&Mathematics with Applications,2010,59(3):1101–1107.
[25]ZHANG W,CAI X,HOLM S.Time-fractional heat equations and negative absolute temperatures[J].Computers and Mathematics with Applications,2014,67(1):164–171.
[26]ZHOU T S,LI C P.Synchronization in fractional-order differential systems[J].Physica D:Nonlinear Phenomena,2005,212(1/2):111–125.
[27]YAN Xiaomei,LIU Ding,GUO Huijun.Chaos control of fractional order Chen system via radial basis function neural network[J].Control Theory&Applications,2010,27(3):344–349.(阎晓妹,刘丁,郭会军.分数阶Chen混沌系统的径向基函数神经滑模控制[J].控制理论与应用,2010,27(3):344–349.)
[28]SUN Meimei,HU Yun’an,WEI Jianming.Adaptive repetitive learning synchronization of uncertain fractional order multi-scroll chaotic systems[J].Control Theory&Applications,2016,33(7):936–944.(孙美美,胡云安,韦建明.不确定分数阶多涡卷混沌系统自适应重复学习同步控制[J].控制理论与应用,2016,33(7):936–944.)
[29]BHALEKAR S,DAFTARDAR-GEJJI V.Synchronization of different fractional order chaotic systems using active control[J].Communications in Nonlinear Science&Numerical Simulation,2010,15(11):3536–3546.
[30]DENG W H.Generalized synchronization in fractional order systems[J].Physical Review E,2007,75(5):49–56.
[31]ADLOO H,ROOPAEI M.Review article on adaptive synchronization of chaotic systems with unknown parameters[J].Nonlinear Dynamics,2011,65(1/2):141–159.
[32]TAN W,WANG Y N,ZENG Z F,et al.Adaptive regulation of uncertain chaos with dynamic neural networks[J].Chinese Physics,2004,13(4):459–463.
[33]ZHANG Ruoxun,YANG Shiping.Designing synchronization schemes for a fractional-order hyperchaotic system[J].Acta Physica Sinica,2008,57(11):6837–6843.(张若洵,杨世平.一个分数阶新超混沌系统的同步[J].物理学报,2008,57(11):6837–6843.)
[34]JIA L X,DAI H,HUI M.Nonlinear feedback synchronisation control between fractional-order and integer-order chaotic systems[J].Chinese Physics B,2010,19(11):194–199.
[35]FU J,YU M,MA T D.Modified impulsive synchronization of fractional order hyperchaotic chen systems[J].Chinese Physics B,2011,20(12):160–166.
[36]BAO Bocheng.Introduction to Chaotic Circuits[M].Beijing:Science Press,2013:46–49.(包伯成.混沌电路导论[M].北京:科学出版社,2013:46–49.)
[37]TAVAZOEI M S,HAERI M.A necessary condition for double scroll attractor existencein fractional-order systems[J].Physics Letters A,2007,367(1):102–113.
[38]ZHOU Ping,WEI Lljia,CHENG Xuefeng.A fractional-order chaotic system and its simulation result by electronic workbench(EWB)[J].Journal of Southwest University,2009,31(1):69–74.(周平,危丽佳,程雪峰.一个分数阶混沌系统及其EWB仿真结果[J].西南大学学报,2009,31(1):69–74.)
[39]MATIGNON D.Stability results for fractional differential equations with applications to control processing[J].Computational Engineering in Systems Applications,1996,2:963–968.
[40]OGATA K.Modern Control Engineering[M].Englewood Cliffs,New Jersey:Prentice-Hall,1990.
[41]LI C P,PENG G J.Chaos in Chen’s system with a fractional order[J].Chaos Solitons&Fractals,2004,22(2):443–450.
[42]ZHANG W W,ZHOU S B,LIAO X F,et al.Estimate the largest Lyapunov exponent of fractional-order systems[C]//International Conference on Communications,Circuits and Systems.Fujian,China:IEEE,2008:1121–1124.
[43]ZHOU P,HUANG K.A new 4-D non-equilibrium fractional-order chaotic system and its circuit implementation[J].Communications in Nonlinear Science&Numerical Simulation,2014,19(6):2005–2011.
[44]KIM H S,EYKHOLT R,SALAS J D.Nonlinear dynamics,delay times,and embedding windows[J].Physica D:Nonlinear Phenomena,1999,127(1/2):48–60.
[45]SATO S,SANO M,SAWADA Y.Practical methods of measuring the generalized dimension and the largest lyapunov exponent in high dimensional chaotic systems[J].Progress of Theoretical Physics,1987,77(1):1–5.
[46]HU Jianbing,HAN Yan,ZHAO Lingdong.A novel stablility theorem for fractional systems and its applying in synchronizing fractional chaotic system based on back-stepping approach[J].Acta Physica Sinica,2009,58(4):2235–2239.(胡建兵,韩焱,赵灵冬.一种新的分数阶系统稳定理论及在backstepping方法同步分数阶混沌系统中的应用[J].物理学报,2009,58(4):2235–2239.)
[47]MA Tiedong,JIANG Weibo,FU Jie,et al.Adaptive synchronization of a class of fractional-order chaotic systems[J].Acta Physica Sinica,2012,61(16):593–598.(马铁东,江伟波,浮洁,等.一类分数阶混沌系统的自适应同步[J].物理学报,2012,61(16):593–598.)