多涡卷混沌吸引子的电路研究
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摘要
本论文介绍了用阶梯波序列产生一维、二维和三维多涡卷混沌吸引子的数学模型,画出了用阶梯波序列产生一维、二维和三维多涡卷混沌吸引子的相应平衡点图形。以此为比较分析了用三角波构造多涡卷混沌系统的混沌动力学特性,总结了用三角波构造多涡卷混沌系统产生多涡卷混沌吸引子的可行性和优点。在电路分析中找到了极值混沌电路、绝对值混沌电路,分段混沌电路和阶梯波混沌系统的相关连的切入点,所以三角波这种与分段电路类同的数学表式在电路搭建上是可以以这几种电路为基础来产生多涡卷混沌吸引子的。在对这些电路进行分岔图和李雅普诺夫指数数值计算的同时对这些电路进行了电路模拟和理论分析。最后电路模拟了用三角波序列产生三维多涡卷混沌吸引子的混沌电路。该混沌电路由积分器,三角波序列发生器和联动转换开关三个部分构成,主要特点是三角波序列的幅度,宽度,平衡点,转折点,斜率等参数可调,从而能产生大小和形状可调的多涡卷混沌吸引子。
This paper introduces the families of scroll grid attractors dynamic characteristic and it’s equilibrium points draw proposes a new approach for generating multi-scroll chaotic attractors via triangular wave series . By the two circuits can produce one variable breakpoint circuit.About these circuits’bifurcation diagram and Lyapunov exponent are received. The most expoment that this key is very simple, the circuit using only resistors,capacitors, and inverting operational amplifiers is described. The chaotic dynamic characteristic of multi-scroll system constructed by triangular wave series is further investigated . A blocking circuit diagram , including integrator, triangular wave series generators, switch linkage, is designed for the hardware implementations. The triangular wave series developed here can adjust the swings,widths, equilibrium points, breakpoints, and slopes so as to generate a large number of scrolls with adjustable sizes and shapes. Moreover, the number of scrolls can be controlled via switch linkage, the simulating demonstrates that this method can be a new approach for generating multi-scroll chaotic attractors
This paper introduces the families of scroll grid attractors dynamic characteristic and it’s equilibrium points draw proposes a new approach for generating multi-scroll chaotic attractors via triangular wave series . By the two circuits can produce one variable breakpoint circuit.About these circuits’bifurcation diagram and Lyapunov exponent are received. The most expoment that this key is very simple, the circuit using only resistors,capacitors, and inverting operational amplifiers is described. The chaotic dynamic characteristic of multi-scroll system constructed by triangular wave series is further investigated . A blocking circuit diagram , including integrator, triangular wave series generators, switch linkage, is designed for the hardware implementations. The triangular wave series developed here can adjust the swings,widths, equilibrium points, breakpoints, and slopes so as to generate a large number of scrolls with adjustable sizes and shapes. Moreover, the number of scrolls can be controlled via switch linkage, the simulating demonstrates that this method can be a new approach for generating multi-scroll chaotic attractors
引文
[1]岳丽娟,陈艳艳,彭建华.用系统变量比例脉冲方法控制超混沌的电路实验研究[J].物理学报,2001,50(11):2097-2102.
[2]周平,罗小华,陈海燕.一个混沌电路及其实验结果[J].物理学报,2005,54(11):5048-5052.
[3]Kataoka,M&Satio,T.A 4- D chaotic oscillator with a hysteresis 2- Port VCCS[J] .IEEE Trans CAS-I,2000,(V):418-421.
[4]Saito,T.An approach toward higher dimensional hysteresis chaos generators[J] .IEEE Trans CAS-I,1990,37(3) :399-409.
[5]Elwakil A S,Salama K N,Kenndey M P.A system for chaos generation and its implementation in monolithic form[J].IEEE Trans CAS-I,2000,(V):217-220.
[6]Kapitaniak T,Chua L O.Hyperchaotic attractors of unidirectionally- coupled Chua’s circuit[J].Int J Bifurc Chaos,1994,4(4):477-482.
[7]Suykens J A K,Curran P F,Chua L O.Robust synthesis for master- slave synchronization of Lur’e Systems[J].IEEE Trans CAS-I,1999,46(7):841-850.
[8]Yalcin M E,Suykens J A K,Vandewalle J.Hyperchaotic n-scroll attractors[J].IEEE NDES,2000,25-28.
[9]孙克辉,牟俊,张泰山.一类离散混沌系统的反馈控制同步方法及条件.广西师范大学学报,2005,23 (1):80-833.
[10]姜德平,罗小曙,孙琳,等.五阶超混沌电路系统的自适应同步研究[J].广西师范大学学报,2005,23 (1):72-75.
[11]陶朝海,陆君安,吕金虎.统一混沌系统的反馈同步[J].物理学报,2002,51 (7):1497-1501.
[12]王铁邦,覃团发,陈光旨.超混沌系统的耦合同步[J] .物理学报,2001,50 (10):1851-1855.
[13]李世华,蔡海兴.Chen 氏混沌电路实现与同步控制实验研究[J].物理学报,2004,53 (6):1687-1673.
[14]吴本科,肖苏,高峰.利用驻波实验研究混沌现象[J].物理实验,2006,26(1):7-10.
[15]刘建东.变型蔡氏电路中混沌控制的实验研究[J].物理实验,2005,25(3):7-10.
[16]何国光,曹志彤.混沌神经网络的控制[J].物理学报,2001,50 (11):2103-2107.
[17]丘水生.混沌吸引子的周期轨道理论研究[J].电路与系统学报,2003,8(6):1-5.
[18]丘水生,陈艳峰.混沌保密通信的若干问题及混沌加密新方案[J].华南理工大学学报,2002,30 (11):75-80.
[19]Liu J F,Chen G R,Zhang S C.DYNAMICAL ANALYSIS OF A NEW CHAOTIC ATTRACTOR[J] .Int J Bifurc Chaos,2002,12(5) :1001-1015.
[20] Liu J H,Chen G R,Cheng D Z.BRINGE THE GAP BETWRRN THE LORENZ SYSTEM AND THE CHENSYSTEM[J].Int J Bifurc Chaos.2002,12(12):2917-2926.
[21]Sprott J C.Simple chaotic systems and circuits [J].Am J Phys,2000,68(8) :758-763.
[22]Sprott J C.A new class of chaotic circuit [J]. Phys Letters A,2000,266(2) :19-23.
[23]刘崇新.蔡氏对偶混沌电路分析[J].物理学报,2002,51(6):1198-1202.
[23]陈菊芳,程丽,刘影,等.延迟变量反馈法控制离散混沌系统的电路实验[J].物理学报,2003,52(1):18-24.
[23]匡锦俞,邓昆,黄荣怀,等.利用时空混沌同步进行数字加密通信[J].物理学报,2001,50(10):1856-1861.
[24] Elwakil A S,Kennedy M P .Imporved implementation of Chua’s chaotic oscillator using current feedback opamp. IEEE Trans CAS-I,2000,47(1) :76-79.
[25] Yalcin M E,Ozoguz S,Suykens J A K,et al .n-Scroll chaos generators:Asimple circuit model[J] . Electron Lett,2000,37(3) :147-148.
[26]Arena P,Baglio S,Fortuna L,et al.Generation of n-double scrolls via cellar neural net-works[J] .Int J Cric Theor Appl,1996,24,241-252.
[27]Suykens J A K,Vandewalle J.Generation of n-double scrolls(n=1,2,3,…)[J] .IEEE Trans CAS-I,1993,40(11) :861-867.
[28] Suykens J A K,Chua L O.n-Double scroll hypercubes in 1-D CNNS[J] .Int J Bifurc Chaos,1997,7(8):1873-1885.
[29]Yalcin M E,Suykens J A K,Vandewalle J.Experimental confirmation of 3-and 5-scroll attractors from a generalized Chua’s circuit[J].IEEE Trans CAS-I,2000,47(3) :425-429.
[30]Tang W K S,Zhong G Q,Chen G,et al.Generation of N-scroll Attractors Via Sine Funtion[J].IEEE Trans CAS-I,48(11):1369-1372.
[31]Zhong G Q , Man k F , Chen G R.A SYSTEMATIC APPROACH TO GENERATING N-SCROLL ATTRACTORS[J] .Int J Bifurc Chaos,2002,12(12):2907-2915.
[32] Yalcin M E,Suykens J A K,Vandewalle J.FAMILIES OF SCROLL GRID ATTRACTORS[J].Int J Bifurcation and Chaos,2002,12(1) :23-41.
[33]禹思敏,林清华,丘水生.四维系统中多涡卷混沌与超混沌吸引子的仿真研究[J].物理学报,2003,52(1):25-33.
[34]禹思敏,林清华,丘水生.一类多折叠环面混沌吸引子[J].物理学报,2004,53(7):2084-2088.
[35]禹思敏.一种新型混沌产生器[J].物理学报,2004,53(12):4111-4119.
[36]刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2005.
[37]陈永泰,马凌.非线性振荡电路的混沌分析[J].实验技术与管理,2006,23(11):39-40.
[38]薛莉,彭建华,张立静.用滤波反馈信号控制时间延迟混沌系统[J].深圳大学学报,2006,23(10):337-341.
[39]王发强,刘崇新.新的变形蔡氏电路及实验[J].通信学报,2006,27(9):103-105.
[40]王发强,刘崇新.Liu 混沌系统的混沌分析及电路实验的研究[J].物理学报,2006,55(10):5061-5069.
[41]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[J].物理学报,2006,55(10):5055-5060.
[2]周平,罗小华,陈海燕.一个混沌电路及其实验结果[J].物理学报,2005,54(11):5048-5052.
[3]Kataoka,M&Satio,T.A 4- D chaotic oscillator with a hysteresis 2- Port VCCS[J] .IEEE Trans CAS-I,2000,(V):418-421.
[4]Saito,T.An approach toward higher dimensional hysteresis chaos generators[J] .IEEE Trans CAS-I,1990,37(3) :399-409.
[5]Elwakil A S,Salama K N,Kenndey M P.A system for chaos generation and its implementation in monolithic form[J].IEEE Trans CAS-I,2000,(V):217-220.
[6]Kapitaniak T,Chua L O.Hyperchaotic attractors of unidirectionally- coupled Chua’s circuit[J].Int J Bifurc Chaos,1994,4(4):477-482.
[7]Suykens J A K,Curran P F,Chua L O.Robust synthesis for master- slave synchronization of Lur’e Systems[J].IEEE Trans CAS-I,1999,46(7):841-850.
[8]Yalcin M E,Suykens J A K,Vandewalle J.Hyperchaotic n-scroll attractors[J].IEEE NDES,2000,25-28.
[9]孙克辉,牟俊,张泰山.一类离散混沌系统的反馈控制同步方法及条件.广西师范大学学报,2005,23 (1):80-833.
[10]姜德平,罗小曙,孙琳,等.五阶超混沌电路系统的自适应同步研究[J].广西师范大学学报,2005,23 (1):72-75.
[11]陶朝海,陆君安,吕金虎.统一混沌系统的反馈同步[J].物理学报,2002,51 (7):1497-1501.
[12]王铁邦,覃团发,陈光旨.超混沌系统的耦合同步[J] .物理学报,2001,50 (10):1851-1855.
[13]李世华,蔡海兴.Chen 氏混沌电路实现与同步控制实验研究[J].物理学报,2004,53 (6):1687-1673.
[14]吴本科,肖苏,高峰.利用驻波实验研究混沌现象[J].物理实验,2006,26(1):7-10.
[15]刘建东.变型蔡氏电路中混沌控制的实验研究[J].物理实验,2005,25(3):7-10.
[16]何国光,曹志彤.混沌神经网络的控制[J].物理学报,2001,50 (11):2103-2107.
[17]丘水生.混沌吸引子的周期轨道理论研究[J].电路与系统学报,2003,8(6):1-5.
[18]丘水生,陈艳峰.混沌保密通信的若干问题及混沌加密新方案[J].华南理工大学学报,2002,30 (11):75-80.
[19]Liu J F,Chen G R,Zhang S C.DYNAMICAL ANALYSIS OF A NEW CHAOTIC ATTRACTOR[J] .Int J Bifurc Chaos,2002,12(5) :1001-1015.
[20] Liu J H,Chen G R,Cheng D Z.BRINGE THE GAP BETWRRN THE LORENZ SYSTEM AND THE CHENSYSTEM[J].Int J Bifurc Chaos.2002,12(12):2917-2926.
[21]Sprott J C.Simple chaotic systems and circuits [J].Am J Phys,2000,68(8) :758-763.
[22]Sprott J C.A new class of chaotic circuit [J]. Phys Letters A,2000,266(2) :19-23.
[23]刘崇新.蔡氏对偶混沌电路分析[J].物理学报,2002,51(6):1198-1202.
[23]陈菊芳,程丽,刘影,等.延迟变量反馈法控制离散混沌系统的电路实验[J].物理学报,2003,52(1):18-24.
[23]匡锦俞,邓昆,黄荣怀,等.利用时空混沌同步进行数字加密通信[J].物理学报,2001,50(10):1856-1861.
[24] Elwakil A S,Kennedy M P .Imporved implementation of Chua’s chaotic oscillator using current feedback opamp. IEEE Trans CAS-I,2000,47(1) :76-79.
[25] Yalcin M E,Ozoguz S,Suykens J A K,et al .n-Scroll chaos generators:Asimple circuit model[J] . Electron Lett,2000,37(3) :147-148.
[26]Arena P,Baglio S,Fortuna L,et al.Generation of n-double scrolls via cellar neural net-works[J] .Int J Cric Theor Appl,1996,24,241-252.
[27]Suykens J A K,Vandewalle J.Generation of n-double scrolls(n=1,2,3,…)[J] .IEEE Trans CAS-I,1993,40(11) :861-867.
[28] Suykens J A K,Chua L O.n-Double scroll hypercubes in 1-D CNNS[J] .Int J Bifurc Chaos,1997,7(8):1873-1885.
[29]Yalcin M E,Suykens J A K,Vandewalle J.Experimental confirmation of 3-and 5-scroll attractors from a generalized Chua’s circuit[J].IEEE Trans CAS-I,2000,47(3) :425-429.
[30]Tang W K S,Zhong G Q,Chen G,et al.Generation of N-scroll Attractors Via Sine Funtion[J].IEEE Trans CAS-I,48(11):1369-1372.
[31]Zhong G Q , Man k F , Chen G R.A SYSTEMATIC APPROACH TO GENERATING N-SCROLL ATTRACTORS[J] .Int J Bifurc Chaos,2002,12(12):2907-2915.
[32] Yalcin M E,Suykens J A K,Vandewalle J.FAMILIES OF SCROLL GRID ATTRACTORS[J].Int J Bifurcation and Chaos,2002,12(1) :23-41.
[33]禹思敏,林清华,丘水生.四维系统中多涡卷混沌与超混沌吸引子的仿真研究[J].物理学报,2003,52(1):25-33.
[34]禹思敏,林清华,丘水生.一类多折叠环面混沌吸引子[J].物理学报,2004,53(7):2084-2088.
[35]禹思敏.一种新型混沌产生器[J].物理学报,2004,53(12):4111-4119.
[36]刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2005.
[37]陈永泰,马凌.非线性振荡电路的混沌分析[J].实验技术与管理,2006,23(11):39-40.
[38]薛莉,彭建华,张立静.用滤波反馈信号控制时间延迟混沌系统[J].深圳大学学报,2006,23(10):337-341.
[39]王发强,刘崇新.新的变形蔡氏电路及实验[J].通信学报,2006,27(9):103-105.
[40]王发强,刘崇新.Liu 混沌系统的混沌分析及电路实验的研究[J].物理学报,2006,55(10):5061-5069.
[41]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[J].物理学报,2006,55(10):5055-5060.