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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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