动态共存吸引子系统动力学性质研究
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摘要
由于动态共存吸引子Newton-Leipnik(N-L)系统,具有上下两个不同的奇异吸引子,所以它的动力学性质更为丰富,潜在应用价值更大。论文从动力学性质分析和电路实现两个方面对N-L系统进行了研究。
理论分析方面,运用庞加莱截面、李亚普诺夫指数、李亚普诺夫维数及分岔图等工具对N-L系统进行了详细的探讨,并结合图形仿真和计算,进一步研究了N-L系统的动力学行为随参数变化的演化特性。
电路实现方面,首先对N-L方程的状态变量进行比例压缩变换、微分-积分转换和时间尺度变换。其次根据变换后的方程设计出各模块电路,再将各模块按方程中各状态变量的对应关系联结起来。结果表明N-L方程由反相加法器、积分器、乘法器和反相器等常用器件即可实现,而且这种实现方式具有如下三方面优点:(1)模块化设计具有普适性,比较容易扩展到其它非线性系统;(2)由于采用了反相加法器,各个电路参数独立可调,互不影响,便于电路参数选择和调试;(3)混沌信号的频谱分布范围可由积分电阻或积分电容大小的改变来实现,便于实际应用。最后,通过EWB电路实现与MATLAB仿真结果的对比表明研究方案的合理性。
Due to the dynamical co-existing attractor, the Newton-Leipnik (N-L) system has two different strange attractors, which may possess much more enriched dynamic properties and potential values. In this paper, dynamical analysis and circuit implementation of N-L system are addressed.
As to theoretical analysis, Poincare section drawing, Lyapunov exponent computation, Lyapunov dimension calculation and bifurcation diagram drawing were borrowed, under auspices of graphic simulation and calculation, the character of dynamics evolution of the N-L system corresponding to parameter perturbation was investigated thoroughly.
As to circuit implementation, variable-scale attenuation, differential to integral conversion and time-scale transform were executed to the N-L equation at first. Then, each module circuit was designed and linked with the other ones according to state variable couplings. And, going further, results demonstrate that the N-L system can be realized by Inverted Adders, Integrators, Inverters and the other easily obtained devices. Scheme supposed here outperforms the other ones in the following three points: (1) Modularization design is universal, and can be easily expanded to the other nonlinear systems. (2) The circuit parameters can be adjusted independently because of usage of Inverted Adder, so it’s convenient for parameter choice and debugging. (3) The chaotic signal spectrum distribution can be regulated by adjustment of the value of the Integral resistor or capacitance, which makes it liable for engineering application. Circuit implementation via EWB and simulation under MATLAB for the N-L system verify the efficacy of the put forward method.
理论分析方面,运用庞加莱截面、李亚普诺夫指数、李亚普诺夫维数及分岔图等工具对N-L系统进行了详细的探讨,并结合图形仿真和计算,进一步研究了N-L系统的动力学行为随参数变化的演化特性。
电路实现方面,首先对N-L方程的状态变量进行比例压缩变换、微分-积分转换和时间尺度变换。其次根据变换后的方程设计出各模块电路,再将各模块按方程中各状态变量的对应关系联结起来。结果表明N-L方程由反相加法器、积分器、乘法器和反相器等常用器件即可实现,而且这种实现方式具有如下三方面优点:(1)模块化设计具有普适性,比较容易扩展到其它非线性系统;(2)由于采用了反相加法器,各个电路参数独立可调,互不影响,便于电路参数选择和调试;(3)混沌信号的频谱分布范围可由积分电阻或积分电容大小的改变来实现,便于实际应用。最后,通过EWB电路实现与MATLAB仿真结果的对比表明研究方案的合理性。
Due to the dynamical co-existing attractor, the Newton-Leipnik (N-L) system has two different strange attractors, which may possess much more enriched dynamic properties and potential values. In this paper, dynamical analysis and circuit implementation of N-L system are addressed.
As to theoretical analysis, Poincare section drawing, Lyapunov exponent computation, Lyapunov dimension calculation and bifurcation diagram drawing were borrowed, under auspices of graphic simulation and calculation, the character of dynamics evolution of the N-L system corresponding to parameter perturbation was investigated thoroughly.
As to circuit implementation, variable-scale attenuation, differential to integral conversion and time-scale transform were executed to the N-L equation at first. Then, each module circuit was designed and linked with the other ones according to state variable couplings. And, going further, results demonstrate that the N-L system can be realized by Inverted Adders, Integrators, Inverters and the other easily obtained devices. Scheme supposed here outperforms the other ones in the following three points: (1) Modularization design is universal, and can be easily expanded to the other nonlinear systems. (2) The circuit parameters can be adjusted independently because of usage of Inverted Adder, so it’s convenient for parameter choice and debugging. (3) The chaotic signal spectrum distribution can be regulated by adjustment of the value of the Integral resistor or capacitance, which makes it liable for engineering application. Circuit implementation via EWB and simulation under MATLAB for the N-L system verify the efficacy of the put forward method.
引文
[1]陈关荣.控制动力系统的分岔现象.控制理论与应用,2001,18(2):153-159
[2] M.P. Kednedy. Chaos in the colpitts oscillator. IEEE Trans. Circuit and system,1994, 41(11):771-774
[3] O. Morgul. Wien bridge based RC chaos generator. Electronics Letters,1995,31(24):2058-2059
[4] A.Namajunas,A.Tamasevicius. Simple RC chaotic oscillator. Electronics Letters,1996,32(11):945-946
[5] Leipnik R B,Newton T A.Double strange attractors in rigid body motion with Linear feedback control.Physics Lettera A,1981,86(2):63-67
[6] Lofaro T.A model of the dynamics of the Newton-Leipnik attractor.International Journal of Bifurcation and Chaos,1997,7(12):2723-2733
[7] Marlin B A.Periodic orbits in the Newton-Leipnik system.International Journal of Bifurcation and Chaos,2002,12(3):511-523
[8] Richter H.Controlling chaotic systems with multiple strange attractors.Physics Letters A,2002,300:182-188
[9] Wang X D,Tian L X.Bifurcation analysis and linear control of the Newton-Leipnik system.Chaos,Solitons and Fractals,2006,27:31-38
[10]蔡国梁,田立新,范兴华.Newton-Leipnik系统的慢流形表达式.江苏大学学报(自然科学版),2005,26(5):405-408
[11]王学弟,田立新,李医民.Newton-Leipnik系统的线性反馈控制与同步研究.江苏大学学报(自然科学版),2004,25(5):417-420
[12]宋运忠,赵光宙,齐冬莲.Passive control of chaotic system with multiple strange attractors. Chinese Physics,2006,15(10):2266-2270
[13]赵光宙,齐冬莲.混沌控制理论及应用.电工技术学报,2001,16(5):77-82
[14] Wolf A,Swoft J.B. Determining Lyapunov exponents from a time series. Phy.D,1985,16D(3):285-317
[15] Parker T S,Chua L O. Practical Numerical Algorithms for Chaotic Systems. NewYork:Spring-Verlag,World Publishing Corp,1989
[16]齐冬莲.连续混沌动力学系统的控制理论研究:[博士学位论文].杭州:浙江大学电气工程学院,2002
[17]黄润生.混沌及其应用.武汉:武汉大学出版社,2000
[18] Benettin G,Galgani L,Strelcyn J.M. Kolmogorov entropy and numerical experiments. Phys.Rev.A,1976,14:2338-2345
[19] Mcdonald E J,Higham D J. Error analysis of QR algorithm for computing Lyapunov exponent. Electronic Transactions on Numerical Analysis,2001,12:234-251
[20] Barana G Tsuda. A new method for computing Lyapunov exponents. Physics Letter A,1993,175:421-427
[21] Rosenstein M T,Collins J J,De Luca C J. A practical methol for calculating largest Lyapunov exponents from small data sets. Physica D,1993,65(1):117-121
[22] Frederickson P ,Kaplan J,Yorke L ,et al. The Lyapunov dimension of strange attractors. Journal of Differential Equation,1983,49:185-207
[23] Yunzhong Song,Guangzhou Zhao, Donglian Qi. Passive control of chaotic system with multiple strange attractors. Chinese Physics, 2006,15(10):2266-2270
[24] Qiang Jia. Chaos control and synchronization of the Newton-Leipnik chaotic system. Chaos, Solitons&Fractals,2008,35(4):814-824
[25] Sheu L J, Chen H K. Chaos in the Newton-Leipnik system with fractional order. Chaos, Solitons&Fractals, 2006,27:1024-1026
[26]李亚,禹思敏,戴青云,等.一种新的蔡氏电路设计方法与硬件实现.物理学报,2006,55(8):3938-3945.
[27]邹艳丽,罗晓曙,陈关荣.用状态变量的一阶反馈法实现蔡电路的混沌控制.控制理论与应用,2004,21(4):565-573
[28]黄秋楠,陈菊芳,彭建华.离散混沌电路的实现.物理实验,2003,23(7):10-22
[29]王发强,刘崇新. Liu混沌系统的线性反馈同步控制及电路实验的研究.物理学报,2006,55(10):5055-5060
[30]黄道平. MATLAB与控制系统的数字仿真及CAD.北京:化学工业出版社,2004
[31]周晓阳.数学实验与Matlab.武汉:华中科技大学出版社,2002
[32]刘文波.混沌工程应用基础研究:[博士论文].南京航空航天大学,2001
[33] E.H.Baghious,P.Jarry. Lorenz attractor from differential equations with piecewise-linear terms. International Journal of Bifurcation and Chaos,1993,3(1):201-210
[34] A.S.Elwakil,M.P.Kennedy. Chua’s circuit decmposition:A systematic design approach for chaotic oscillators. Journal of the Franklin Institute,2000,337:251-26
[2] M.P. Kednedy. Chaos in the colpitts oscillator. IEEE Trans. Circuit and system,1994, 41(11):771-774
[3] O. Morgul. Wien bridge based RC chaos generator. Electronics Letters,1995,31(24):2058-2059
[4] A.Namajunas,A.Tamasevicius. Simple RC chaotic oscillator. Electronics Letters,1996,32(11):945-946
[5] Leipnik R B,Newton T A.Double strange attractors in rigid body motion with Linear feedback control.Physics Lettera A,1981,86(2):63-67
[6] Lofaro T.A model of the dynamics of the Newton-Leipnik attractor.International Journal of Bifurcation and Chaos,1997,7(12):2723-2733
[7] Marlin B A.Periodic orbits in the Newton-Leipnik system.International Journal of Bifurcation and Chaos,2002,12(3):511-523
[8] Richter H.Controlling chaotic systems with multiple strange attractors.Physics Letters A,2002,300:182-188
[9] Wang X D,Tian L X.Bifurcation analysis and linear control of the Newton-Leipnik system.Chaos,Solitons and Fractals,2006,27:31-38
[10]蔡国梁,田立新,范兴华.Newton-Leipnik系统的慢流形表达式.江苏大学学报(自然科学版),2005,26(5):405-408
[11]王学弟,田立新,李医民.Newton-Leipnik系统的线性反馈控制与同步研究.江苏大学学报(自然科学版),2004,25(5):417-420
[12]宋运忠,赵光宙,齐冬莲.Passive control of chaotic system with multiple strange attractors. Chinese Physics,2006,15(10):2266-2270
[13]赵光宙,齐冬莲.混沌控制理论及应用.电工技术学报,2001,16(5):77-82
[14] Wolf A,Swoft J.B. Determining Lyapunov exponents from a time series. Phy.D,1985,16D(3):285-317
[15] Parker T S,Chua L O. Practical Numerical Algorithms for Chaotic Systems. NewYork:Spring-Verlag,World Publishing Corp,1989
[16]齐冬莲.连续混沌动力学系统的控制理论研究:[博士学位论文].杭州:浙江大学电气工程学院,2002
[17]黄润生.混沌及其应用.武汉:武汉大学出版社,2000
[18] Benettin G,Galgani L,Strelcyn J.M. Kolmogorov entropy and numerical experiments. Phys.Rev.A,1976,14:2338-2345
[19] Mcdonald E J,Higham D J. Error analysis of QR algorithm for computing Lyapunov exponent. Electronic Transactions on Numerical Analysis,2001,12:234-251
[20] Barana G Tsuda. A new method for computing Lyapunov exponents. Physics Letter A,1993,175:421-427
[21] Rosenstein M T,Collins J J,De Luca C J. A practical methol for calculating largest Lyapunov exponents from small data sets. Physica D,1993,65(1):117-121
[22] Frederickson P ,Kaplan J,Yorke L ,et al. The Lyapunov dimension of strange attractors. Journal of Differential Equation,1983,49:185-207
[23] Yunzhong Song,Guangzhou Zhao, Donglian Qi. Passive control of chaotic system with multiple strange attractors. Chinese Physics, 2006,15(10):2266-2270
[24] Qiang Jia. Chaos control and synchronization of the Newton-Leipnik chaotic system. Chaos, Solitons&Fractals,2008,35(4):814-824
[25] Sheu L J, Chen H K. Chaos in the Newton-Leipnik system with fractional order. Chaos, Solitons&Fractals, 2006,27:1024-1026
[26]李亚,禹思敏,戴青云,等.一种新的蔡氏电路设计方法与硬件实现.物理学报,2006,55(8):3938-3945.
[27]邹艳丽,罗晓曙,陈关荣.用状态变量的一阶反馈法实现蔡电路的混沌控制.控制理论与应用,2004,21(4):565-573
[28]黄秋楠,陈菊芳,彭建华.离散混沌电路的实现.物理实验,2003,23(7):10-22
[29]王发强,刘崇新. Liu混沌系统的线性反馈同步控制及电路实验的研究.物理学报,2006,55(10):5055-5060
[30]黄道平. MATLAB与控制系统的数字仿真及CAD.北京:化学工业出版社,2004
[31]周晓阳.数学实验与Matlab.武汉:华中科技大学出版社,2002
[32]刘文波.混沌工程应用基础研究:[博士论文].南京航空航天大学,2001
[33] E.H.Baghious,P.Jarry. Lorenz attractor from differential equations with piecewise-linear terms. International Journal of Bifurcation and Chaos,1993,3(1):201-210
[34] A.S.Elwakil,M.P.Kennedy. Chua’s circuit decmposition:A systematic design approach for chaotic oscillators. Journal of the Franklin Institute,2000,337:251-26