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在Chen-Lai算法和Wang-Chen算法中, 模函数的定义域均为(-∞, +∞). 然而, 在电子电路等技术实现中, 模函数定义在有限区域上则更符合实际情况. 本文以有限区域条件下模函数为正弦函数的离散时间系统反控制为典型实例,给出了受控系统在Li-Yorke意义下混沌的充分条件和严格的理论证明,从而能根据定理给出的充分条件和器件自身规定的一个有限区域或动态范围的约束条件来共同确定电路的具体参数范围,为电路设计与技术实现提供理论依据.基于这一方法, 设计了有限区域条件下模函数为正弦函数的离散时间系统反控制电路,给出了电路实验结果,证实了本方法的可行性. 本文的这种方法也可用于解决有限区域条件下模函数为其他非线性函数的离散时间反控制与电路实现问题.In the Chen-Lai and Wang-Chen algorithm , the modular functions are both defined in (-∞,+∞). A modular function, however, in the implementation of electronic circuit, is more reasonable in line with the actual situation if it is defined in a finite region. We take for example the anti-control of a discrete time system, of which the modular function is sine function on the basis of a finite region. And in the sense of Li-Yorke, the chaotic sufficient condition and the rigorous theory proof are provided. As a result, ranges of specific circuit parameters can be determined by both the sufficient conditions resulting from the theorem, and a finite region defined by the device, or the constraint conditions of a dynamic range. Therefore, this provides a fundamental basis for the circuit design and its technology. Based on this method, the anti-control circuit of the discrete time system is designed, of which the modular function is sine function in a finite region. And the experimental results are given for confirming the feasibility of the method. The method presented in this paper can also be applied to the circuit implementation and the anti-control of a discrete time system, of which the modular function is other nonlinear function.
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Keywords:
- limited regional conditions /
- discrete-time systems /
- anti-control /
- circuit implementation
[1] Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196
[2] Chen G, Dong X 1998 From Chaos to Order: Methodologies, Perspectives and Applications (World Scientific Pub. Co., Singapore) p26
[3] Chen G 1999 Controlling chaos and Bifurcations in Engineering Systems (CRC Press, USA) p38
[4] Chen G, Yu X 2003 Chaos Control:Theory and Applications (Springer-Verlag, Berlin) p79
[5] Schiff S J, Jerger K, Duong D H, Chang T, Spano M L, Ditto W L 1994 Nature 363 411
[6] Chen G, Lai D 1996 Int. J. of Bifurc. Chaos 6 1341
[7] Chen G, Lai D 1997 Proc. IEEE Conf. Deci. control 1997 SanDiego, CA, Dec 10-12 ,367
[8] Chen G, Lai D 1998 Int. J. of Bifur. Chaos 8 1585
[9] Oseledec V I 1968 Trans. Moscow Math. Soc. 19 197
[10] Devaney R L 1987 An Introduction to Chaotic Dynamical Systems (New York: Addison-Wesley) 10-15
[11] Li T Y, Yorke J A 1975 Ameri. Math. Monthly 82 481
[12] Wang X F, Chen G R 2000IEEE Trans. on Circcuits Syst. I 47 410
[13] Chen G R, Wang X F 2006 Chaotification of Dynamical Systems-Theory, Method and Applications (Shanghai: Shanghai Jiaotong University Press) p18-21 (in Chinese) [陈关荣, 汪小帆 2006 动力系统的混沌化—理论、方法与应用 (上海: 上海交通大学出版社) 第18—31页]
[14] Chen J F, Cheng L, Liu Y, Peng J H 2003 Acta Phys. Sin. 52 3290 (in Chinese) [陈菊芳, 程丽, 刘颖, 彭建华 2003 物理学报 52 3290]
[15] Zhang X M, Wang H, Peng J H 2010 J. Shenzhen Univ. Sci. Engi. 27 317 ( in Chinese) [张晓明, 王赫, 彭建华 2010 深圳大学学报理工版 27 317]
[16] Chen J F, Zhang R Y, Peng J H 2003 Acta Phys. Sin. 52 1589 (in Chinese) [陈菊芳, 张入元, 彭建华 2003 物理学报 52 1589]
[17] Feng C W, Cai L, Kang Q ,Peng W D,Bai P,Wang J F 2011 Acta Phys. Sin. 60 110502 (in Chinese) [冯朝文, 蔡理, 康强, 彭卫东, 柏鹏, 王甲富 2011 物理学报 60 110502]
[18] Wang X M, Li Y F 2010 J. Jilin Univ. (Infor. Sci. Edi.) 29 36 (in Chinese) [王学明, 李原福 吉林大学学报(信息科学版) 29 36]
[19] Chen X, Qiu S S 2010Acta Phys. Sin. 59 7630 (in Chinese) [陈旭, 丘水生 2010 物理学报 59 7630]
[20] Marotto F R 1978 J. Math. Anal. Appl. 63 199
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[1] Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196
[2] Chen G, Dong X 1998 From Chaos to Order: Methodologies, Perspectives and Applications (World Scientific Pub. Co., Singapore) p26
[3] Chen G 1999 Controlling chaos and Bifurcations in Engineering Systems (CRC Press, USA) p38
[4] Chen G, Yu X 2003 Chaos Control:Theory and Applications (Springer-Verlag, Berlin) p79
[5] Schiff S J, Jerger K, Duong D H, Chang T, Spano M L, Ditto W L 1994 Nature 363 411
[6] Chen G, Lai D 1996 Int. J. of Bifurc. Chaos 6 1341
[7] Chen G, Lai D 1997 Proc. IEEE Conf. Deci. control 1997 SanDiego, CA, Dec 10-12 ,367
[8] Chen G, Lai D 1998 Int. J. of Bifur. Chaos 8 1585
[9] Oseledec V I 1968 Trans. Moscow Math. Soc. 19 197
[10] Devaney R L 1987 An Introduction to Chaotic Dynamical Systems (New York: Addison-Wesley) 10-15
[11] Li T Y, Yorke J A 1975 Ameri. Math. Monthly 82 481
[12] Wang X F, Chen G R 2000IEEE Trans. on Circcuits Syst. I 47 410
[13] Chen G R, Wang X F 2006 Chaotification of Dynamical Systems-Theory, Method and Applications (Shanghai: Shanghai Jiaotong University Press) p18-21 (in Chinese) [陈关荣, 汪小帆 2006 动力系统的混沌化—理论、方法与应用 (上海: 上海交通大学出版社) 第18—31页]
[14] Chen J F, Cheng L, Liu Y, Peng J H 2003 Acta Phys. Sin. 52 3290 (in Chinese) [陈菊芳, 程丽, 刘颖, 彭建华 2003 物理学报 52 3290]
[15] Zhang X M, Wang H, Peng J H 2010 J. Shenzhen Univ. Sci. Engi. 27 317 ( in Chinese) [张晓明, 王赫, 彭建华 2010 深圳大学学报理工版 27 317]
[16] Chen J F, Zhang R Y, Peng J H 2003 Acta Phys. Sin. 52 1589 (in Chinese) [陈菊芳, 张入元, 彭建华 2003 物理学报 52 1589]
[17] Feng C W, Cai L, Kang Q ,Peng W D,Bai P,Wang J F 2011 Acta Phys. Sin. 60 110502 (in Chinese) [冯朝文, 蔡理, 康强, 彭卫东, 柏鹏, 王甲富 2011 物理学报 60 110502]
[18] Wang X M, Li Y F 2010 J. Jilin Univ. (Infor. Sci. Edi.) 29 36 (in Chinese) [王学明, 李原福 吉林大学学报(信息科学版) 29 36]
[19] Chen X, Qiu S S 2010Acta Phys. Sin. 59 7630 (in Chinese) [陈旭, 丘水生 2010 物理学报 59 7630]
[20] Marotto F R 1978 J. Math. Anal. Appl. 63 199
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