复合复映射M-J集和一类混沌系统的投影同步
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摘要
非线性理论是描述具有无规则结构的复杂系统结构形态的一门新兴边缘科学。它包含了分形、混沌和孤子这三个非常重要的概念。本文侧重研究了分形学中具有重要意义的复合复映射M-J集(简称M-J集)的相关理论方法和一类混沌系统的投影同步问题，并取得了一些重要的研究成果。
(1) 将复合复映射z←(z~2+c)~2+c推广为z←(z~α+c)~β+c(α，β∈R)，研究了由该映射所构造的广义J集的分形结构和裂变演化规律，通过研究发现：α为整数的广义J集具有旋转对称性；α和β为小数时，相角θ主值范围的不同选取，将导致广义J集的不同演化；计算了广义J集之间的Hausdorff距离，这对研究广义J集之间的匹配程度提供了一种新的研究方法。
(2) 研究了由复合复映射z←(z~α+c)~β+c(α，β∈R)所构造的准广义Mandelbbrot集(简称M集)。通过对准广义M集分形图的绘制发现：当α和β为整数时，准广义M集关于x轴对称，并且α和β为为奇数时，准广义M集不仅关于x轴对称，还关于y轴对称，并在理论上给出了证明；利用DeMoivre理论，当α和β为小数时，对相角θ的主值范围的不同选取，准广义M集将出现错动和断裂，出现了雏瓣和部分卫星群。最后将Hausdorff距离应用到准广义M集，得出了不同准广义M集之间的匹配程度。
(3) 实现了一种新混沌系统——变形耦合发电机系统的投影同步。利用状态观测器理论，设计了与该混沌系统相应的投影同步观测器，利用该观测器理论上使得变形耦合发电机系统达到投影同步。最后，对变形耦合发电机系统进行了数值仿真，仿真结果进一步验证了该方法的有效性。
The nonlinear theory is a new developing frontier science which describes the complex systematic structure shape which has a random structure. The nonlinear theory contains three important concepts: Fractal, Chaos and Soliton. Here we lay a particular emphasis on the studying of the theory and methods of the M-J sets (abbreviated to M-J sets) of Compound complex mapping and Projective Synchronization of a class of chaos System. Also we will give out some important research results.
Scholars have made deep researches on the Mandelbrot-Julia sets generated from the complex mapping z← z~α + c(α∈R). Entwistle had studied the J sets of compound complex mapping z←(z~2 +c)~2+c. We generalize complex mapping as z←(z~α+ c)~β+ c(α,β∈R), and research on the structure topological inflexibility and the discontinuity evolution law of the generalized J sets of this mapping. The researches as below: generalized J sets have α-fold rotation symmetry and its center is the origin when α is integer; the different choices of angle lead to the different evolution of generalized J sets; computer the Hausdorff distance between two generalized J sets.
Then we give a research into the generalized M sets of z←(z~α+c)~β+ c(α,β∈R), By plotting the Quasi-M sets, we find that: when α and β are integers, the Quasi-M sets are symmetrical about x axis, when α and β are odd, the Quasi-M sets are symmetrical about x and y axis, this has been illustrated theoretically; Making using of DeMoivre theory, when α and β are decimal fractions, the different choices of angle 9 lead to the different evolution of generalized M sets. Finally, Hausdorff distance is applied to the Quasi-M sets, offer the mathing of two Quasi-M sets.
Finally, we study projective sysnchronization of a new chaos system—a modified coupled dynamos system. Using techniques from the state observer design, we present a systematic, design procedure to synchronize a modified coupled dynamos system by a scaling factor., and have proved the validity of the design. Finally, feasibility of the technique is illustrated for a modified coupled dynamos system.
(1) 将复合复映射z←(z~2+c)~2+c推广为z←(z~α+c)~β+c(α，β∈R)，研究了由该映射所构造的广义J集的分形结构和裂变演化规律，通过研究发现：α为整数的广义J集具有旋转对称性；α和β为小数时，相角θ主值范围的不同选取，将导致广义J集的不同演化；计算了广义J集之间的Hausdorff距离，这对研究广义J集之间的匹配程度提供了一种新的研究方法。
(2) 研究了由复合复映射z←(z~α+c)~β+c(α，β∈R)所构造的准广义Mandelbbrot集(简称M集)。通过对准广义M集分形图的绘制发现：当α和β为整数时，准广义M集关于x轴对称，并且α和β为为奇数时，准广义M集不仅关于x轴对称，还关于y轴对称，并在理论上给出了证明；利用DeMoivre理论，当α和β为小数时，对相角θ的主值范围的不同选取，准广义M集将出现错动和断裂，出现了雏瓣和部分卫星群。最后将Hausdorff距离应用到准广义M集，得出了不同准广义M集之间的匹配程度。
(3) 实现了一种新混沌系统——变形耦合发电机系统的投影同步。利用状态观测器理论，设计了与该混沌系统相应的投影同步观测器，利用该观测器理论上使得变形耦合发电机系统达到投影同步。最后，对变形耦合发电机系统进行了数值仿真，仿真结果进一步验证了该方法的有效性。
The nonlinear theory is a new developing frontier science which describes the complex systematic structure shape which has a random structure. The nonlinear theory contains three important concepts: Fractal, Chaos and Soliton. Here we lay a particular emphasis on the studying of the theory and methods of the M-J sets (abbreviated to M-J sets) of Compound complex mapping and Projective Synchronization of a class of chaos System. Also we will give out some important research results.
Scholars have made deep researches on the Mandelbrot-Julia sets generated from the complex mapping z← z~α + c(α∈R). Entwistle had studied the J sets of compound complex mapping z←(z~2 +c)~2+c. We generalize complex mapping as z←(z~α+ c)~β+ c(α,β∈R), and research on the structure topological inflexibility and the discontinuity evolution law of the generalized J sets of this mapping. The researches as below: generalized J sets have α-fold rotation symmetry and its center is the origin when α is integer; the different choices of angle lead to the different evolution of generalized J sets; computer the Hausdorff distance between two generalized J sets.
Then we give a research into the generalized M sets of z←(z~α+c)~β+ c(α,β∈R), By plotting the Quasi-M sets, we find that: when α and β are integers, the Quasi-M sets are symmetrical about x axis, when α and β are odd, the Quasi-M sets are symmetrical about x and y axis, this has been illustrated theoretically; Making using of DeMoivre theory, when α and β are decimal fractions, the different choices of angle 9 lead to the different evolution of generalized M sets. Finally, Hausdorff distance is applied to the Quasi-M sets, offer the mathing of two Quasi-M sets.
Finally, we study projective sysnchronization of a new chaos system—a modified coupled dynamos system. Using techniques from the state observer design, we present a systematic, design procedure to synchronize a modified coupled dynamos system by a scaling factor., and have proved the validity of the design. Finally, feasibility of the technique is illustrated for a modified coupled dynamos system.
引文
[1] M B B. The fractal geometry of nature. San Francisco: Freeman W H, 1982.
[2] 王兴元．广义M-J集的分形机理．大连：大连理工大学出版社，2002．
[3] Peitgen H O, Saupe D. The science of fractal images. Berlin: Springer-Verlag, 1988.
[4] Barnsley M F. Fractals everywhere. Boston: Academic Press Professional, 1993.
[5] 曾文曲，王向阳．分形理论与分形的计算机模拟．沈阳：东北大学出版社，2001．
[6] Li T Y, Yorke J A. Period three implies chaos. American Mathematical Monthly. 1975, 82(10):985-992.
[7] 王兴元．复杂非线性系统中的混沌．北京：电子工业出版社，2003．
[8] Lorenz E N. Deterministic noperodic flow. Journal of the Atmospheric Sciences. 1963, 20(1): 130-141.
[9] May R M. Simple mathematical models with very complicated dynamics. Nature. 1976, 261(3):459-467.
[10] Feigenbaum M J. Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics. 1978, 19(1): 25-52.
[11] 郝柏林．分岔、混沌、奇怪吸引子、湍流及其它．物理学进展．1983，3(3)：329-416．
[12] Eckmann J P. Roads to turbulence in dissipative dynamics system. Reviews of Modem Physics. 1981,53(4): 643-649.
[13] Brandstater A, Swinney H L. Strange attractors in weakly turbulent Couette-Taylor flow. Physical Review A. 1987, 35(5): 2207-2219.
[14] 王兴元．神经元网络奇怪吸引子的计算机模拟．计算机研究与发展．2001，38(6)：668-673．
[15] 王兴元，朱伟勇．三维奇怪吸引子透视图的计算机模拟．东北大学学报．1999，20(1)：22-24．
[16] Glynn E F. The evolution of the Gingerbread Man. Computers and Graphics. 1991, 15(4): 579-582.
[17] Sasmor J C. Fractals for functions with rational exponent. Computers and Graphics. 2004, 28(4):601-615.
[18] Lakhtakia A. On the symmetries of the J sets for the process z←z~p+c. Journal of Physics A: Mathematical and General. 1987, 20(8): 3533-3535.
[19] Gujar U G, Bhavsar V C. Fractals from z←z~α+c in the Complex c-plane. Computers and Graphics. 1991, 15(3): 441-449.
[20] Gujar U G, Bhavsar V C, Vangala N. Fractals images from z←z~α+c in the complex z-plane. Computers and Graphics. 1992, 16(1): 45-49.
[21] WangX Y, Liu X D, Zhu W Y, et al. Analysis of c-plane fractal images from z←z~α+c for α＜0. Fractals. 2000, 8(3): 307-314.
[22] Romera M, Pastor G, Alvarez G, et al. External arguments of Douady cauliflowers in the M set. Computers and Graphics. 2004, 28(3): 437-449.
[23] Pastor G, Romera M, Alvarez G, et al. Chaotic bands in the M set. Computers and Graphics. 2004, 28(5): 779-784.
[24] Geum Y H, Kim Y I. Accurate computation of component centers in the degree-n bifurcation set. Computers and Mathematics with Applications. 2004, 48(1-2): 163-175.
[25] Wang X Y, Chang P J. Research on fractal structure of generalized M-J sets utilized Lyapunov exponents and periodic scanning techniques. Applied Mathematics and Computation. 2006, 175(2): 1007-1025.
[26] Lakhtakia A. J sets of switched processes. Computers and Graphics. 1991, 15(4): 597-599.
[27] Michelitsch M, Rossler O E. The "Burning ship" and its Quasi-J set. Computers and Graphics. 1992, 16(4): 435-438.
[28] Shirriff K. Fractals from simple polynomial composite functions. Computers and Graphics. 1993, 17(6): 701-703.
[29] Chen Ning, Zhu Weiyong. Bud-sequence conjecture on M fractal image and M-J conjecture between C and Z planes from z←z~w+c(w=α+iβ). Computers and Graphics. 1998, 22(4): 537-546.
[30] Wang X Y. Generalized Mandelbort sets from a class complex mapping system. Applied Mathematics and Computation. 2006, 175(2): 1484-1494.
[31] Wang X Y, Luo C. Generalized J sets from a non-analytic complex mapping. Applied Mathematics and Computation. 2006, 181(1): 113-122.
[32] Entwistle I D. J set art and fractals in the complex plane. Computers and Graphics. 1989, 13(3): 389-392.
[33] Blancharel P. Complex analytic dynamics on the Riemann sphere. Bulletin of the american mathematical society. 1984, 11(11): 88-144.
[34] Pecora L M, Carroll T L. Synchronization in chaotic systems. Physical Review Letters. 1990, 64(8): 821-827.
[35] Carroll T L, Pecora L M. Synchronizing chaotic circuits. IEEE Transactions on Circuits and Systems. 1991, 38(4): 453-456.
[36] Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific, 1998.
[37] 王光瑞，于熙龄，陈式刚．混沌的控制、同步与利用．北京：国防工业出版社，2001．
[38] Agiza H N, Yassen M T. Synchronization of Rossler and Chen chaotic dynamical systems using active control. Physics Letters A. 2000, 278(1): 191-197.
[39] Morgul O, Solak E. Observer based synchronization of chaotic systems. Physical Review E. 1996, 54(5): 4803-4811.
[40] Grassi G, Mascolo S. Nonlinear observer design to synchronize hyperchaotic systems via ascalar signal. IEEE Transactions. Circuits Systems. 1997, 44(10): 1011-1014.
[41] Feki M, Robert B. Secure digital communication using discrete-time chaos synchronization. Chaos, Solitons Fractals. 2003, 18(4): 881-890.
[42] Jiang G P, Tang K S, Chen G. A simple global synchronization criterion for coupled chaotic systems. Chaos, Solitons Fractals. 2003, 15(5): 925-935.
[43] Feki M. An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons and Fractals. 2003, 18(1): 141-148.
[44] Parmananda P. Synchronization using linear and nonlinear feedbacks: a comparison. Physics Letters A. 1998, 240(1-2): 55-59.
[45] Feki M. Observer-based exact synchronization of ideal and mismatched chaotic systems. Physics Letters A. 2003, 309(1-2): 53-60.
[46] Huang L L, Feng R P, Wang M. Synchronization of chaotic systems via nonlinear control. Physics. Letters A. 2004,320(4): 271-275.
[47] Chen H K. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons Fractals. 2005, 23(4): 1245-1251.
[48] Chen M Y, Zhou D H, Shang Y. Synchronizing a class of uncertain chaotic systems. Physics Letters A. 2005. 337(4-6): 384-390.
[49] Shinbrot T, Grebogi C, Ott E, et al. Using small perturbations to control chaos. Nature. 1993, 363(3): 411-417.
[50] Michael G R, Arkady S P, Jurgen K. From phase to lag synchronization in coupled chaotic oscillators. Physical Review Letters. 1997, 78(22): 4193-4196.
[51] Yu X, Song Y. Chaos synchronization via controlling partial state of chaotic systems. International Journal of Bifurcation and chaos. 2001, 11(6): 1737-1741.
[52] Yang X S. On the existence of generalized synchronizor in unidirectionally coupled systems. Applied Mathematics and Computation. 2001,122(1): 71-79.
[53] Ho M C, Hung Y C, Chou C H. Phase and anti-phase synchronization of two chaotic systems by using active control. Physics Letters A. 2002,296(1): 43-48.
[54] Shahverdiev E M, Sivaprakasam S, Shore K A. Lag synchronization in time-delayed systems. Physics Letters A. 2002, 292(6): 320-324.
[55] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Physical Review Letters. 1999, 82(15-12): 3042-3045.
[56] Xu D L, Li Z. Controlled projective synchronization in nonpartially-linear chaotic systems. International Journal of Bifurcation and chaos. 2002, 12(6): 1395-1402.
[57] Kim C M, Rim S H, Key W. Anti-synchronization of chaotic oscillators. Physics Letters A. 2003, 320(1): 39-46.
[58] Hu J, Chen S H, Chen L. Adaptive control for anti-synchronization of Chua's chaotic system. Physics Letters A. 2005,339(6): 455-460.
[59] Wen G L, Xu D L. Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems. Chaos, Solitons Fractals. 2005,26(1): 71-77.
[60] Paraskevopoulos P N. Modern control engineering. New York: Marcel Dekker, 2002.
[2] 王兴元．广义M-J集的分形机理．大连：大连理工大学出版社，2002．
[3] Peitgen H O, Saupe D. The science of fractal images. Berlin: Springer-Verlag, 1988.
[4] Barnsley M F. Fractals everywhere. Boston: Academic Press Professional, 1993.
[5] 曾文曲，王向阳．分形理论与分形的计算机模拟．沈阳：东北大学出版社，2001．
[6] Li T Y, Yorke J A. Period three implies chaos. American Mathematical Monthly. 1975, 82(10):985-992.
[7] 王兴元．复杂非线性系统中的混沌．北京：电子工业出版社，2003．
[8] Lorenz E N. Deterministic noperodic flow. Journal of the Atmospheric Sciences. 1963, 20(1): 130-141.
[9] May R M. Simple mathematical models with very complicated dynamics. Nature. 1976, 261(3):459-467.
[10] Feigenbaum M J. Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics. 1978, 19(1): 25-52.
[11] 郝柏林．分岔、混沌、奇怪吸引子、湍流及其它．物理学进展．1983，3(3)：329-416．
[12] Eckmann J P. Roads to turbulence in dissipative dynamics system. Reviews of Modem Physics. 1981,53(4): 643-649.
[13] Brandstater A, Swinney H L. Strange attractors in weakly turbulent Couette-Taylor flow. Physical Review A. 1987, 35(5): 2207-2219.
[14] 王兴元．神经元网络奇怪吸引子的计算机模拟．计算机研究与发展．2001，38(6)：668-673．
[15] 王兴元，朱伟勇．三维奇怪吸引子透视图的计算机模拟．东北大学学报．1999，20(1)：22-24．
[16] Glynn E F. The evolution of the Gingerbread Man. Computers and Graphics. 1991, 15(4): 579-582.
[17] Sasmor J C. Fractals for functions with rational exponent. Computers and Graphics. 2004, 28(4):601-615.
[18] Lakhtakia A. On the symmetries of the J sets for the process z←z~p+c. Journal of Physics A: Mathematical and General. 1987, 20(8): 3533-3535.
[19] Gujar U G, Bhavsar V C. Fractals from z←z~α+c in the Complex c-plane. Computers and Graphics. 1991, 15(3): 441-449.
[20] Gujar U G, Bhavsar V C, Vangala N. Fractals images from z←z~α+c in the complex z-plane. Computers and Graphics. 1992, 16(1): 45-49.
[21] WangX Y, Liu X D, Zhu W Y, et al. Analysis of c-plane fractal images from z←z~α+c for α＜0. Fractals. 2000, 8(3): 307-314.
[22] Romera M, Pastor G, Alvarez G, et al. External arguments of Douady cauliflowers in the M set. Computers and Graphics. 2004, 28(3): 437-449.
[23] Pastor G, Romera M, Alvarez G, et al. Chaotic bands in the M set. Computers and Graphics. 2004, 28(5): 779-784.
[24] Geum Y H, Kim Y I. Accurate computation of component centers in the degree-n bifurcation set. Computers and Mathematics with Applications. 2004, 48(1-2): 163-175.
[25] Wang X Y, Chang P J. Research on fractal structure of generalized M-J sets utilized Lyapunov exponents and periodic scanning techniques. Applied Mathematics and Computation. 2006, 175(2): 1007-1025.
[26] Lakhtakia A. J sets of switched processes. Computers and Graphics. 1991, 15(4): 597-599.
[27] Michelitsch M, Rossler O E. The "Burning ship" and its Quasi-J set. Computers and Graphics. 1992, 16(4): 435-438.
[28] Shirriff K. Fractals from simple polynomial composite functions. Computers and Graphics. 1993, 17(6): 701-703.
[29] Chen Ning, Zhu Weiyong. Bud-sequence conjecture on M fractal image and M-J conjecture between C and Z planes from z←z~w+c(w=α+iβ). Computers and Graphics. 1998, 22(4): 537-546.
[30] Wang X Y. Generalized Mandelbort sets from a class complex mapping system. Applied Mathematics and Computation. 2006, 175(2): 1484-1494.
[31] Wang X Y, Luo C. Generalized J sets from a non-analytic complex mapping. Applied Mathematics and Computation. 2006, 181(1): 113-122.
[32] Entwistle I D. J set art and fractals in the complex plane. Computers and Graphics. 1989, 13(3): 389-392.
[33] Blancharel P. Complex analytic dynamics on the Riemann sphere. Bulletin of the american mathematical society. 1984, 11(11): 88-144.
[34] Pecora L M, Carroll T L. Synchronization in chaotic systems. Physical Review Letters. 1990, 64(8): 821-827.
[35] Carroll T L, Pecora L M. Synchronizing chaotic circuits. IEEE Transactions on Circuits and Systems. 1991, 38(4): 453-456.
[36] Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific, 1998.
[37] 王光瑞，于熙龄，陈式刚．混沌的控制、同步与利用．北京：国防工业出版社，2001．
[38] Agiza H N, Yassen M T. Synchronization of Rossler and Chen chaotic dynamical systems using active control. Physics Letters A. 2000, 278(1): 191-197.
[39] Morgul O, Solak E. Observer based synchronization of chaotic systems. Physical Review E. 1996, 54(5): 4803-4811.
[40] Grassi G, Mascolo S. Nonlinear observer design to synchronize hyperchaotic systems via ascalar signal. IEEE Transactions. Circuits Systems. 1997, 44(10): 1011-1014.
[41] Feki M, Robert B. Secure digital communication using discrete-time chaos synchronization. Chaos, Solitons Fractals. 2003, 18(4): 881-890.
[42] Jiang G P, Tang K S, Chen G. A simple global synchronization criterion for coupled chaotic systems. Chaos, Solitons Fractals. 2003, 15(5): 925-935.
[43] Feki M. An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons and Fractals. 2003, 18(1): 141-148.
[44] Parmananda P. Synchronization using linear and nonlinear feedbacks: a comparison. Physics Letters A. 1998, 240(1-2): 55-59.
[45] Feki M. Observer-based exact synchronization of ideal and mismatched chaotic systems. Physics Letters A. 2003, 309(1-2): 53-60.
[46] Huang L L, Feng R P, Wang M. Synchronization of chaotic systems via nonlinear control. Physics. Letters A. 2004,320(4): 271-275.
[47] Chen H K. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons Fractals. 2005, 23(4): 1245-1251.
[48] Chen M Y, Zhou D H, Shang Y. Synchronizing a class of uncertain chaotic systems. Physics Letters A. 2005. 337(4-6): 384-390.
[49] Shinbrot T, Grebogi C, Ott E, et al. Using small perturbations to control chaos. Nature. 1993, 363(3): 411-417.
[50] Michael G R, Arkady S P, Jurgen K. From phase to lag synchronization in coupled chaotic oscillators. Physical Review Letters. 1997, 78(22): 4193-4196.
[51] Yu X, Song Y. Chaos synchronization via controlling partial state of chaotic systems. International Journal of Bifurcation and chaos. 2001, 11(6): 1737-1741.
[52] Yang X S. On the existence of generalized synchronizor in unidirectionally coupled systems. Applied Mathematics and Computation. 2001,122(1): 71-79.
[53] Ho M C, Hung Y C, Chou C H. Phase and anti-phase synchronization of two chaotic systems by using active control. Physics Letters A. 2002,296(1): 43-48.
[54] Shahverdiev E M, Sivaprakasam S, Shore K A. Lag synchronization in time-delayed systems. Physics Letters A. 2002, 292(6): 320-324.
[55] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Physical Review Letters. 1999, 82(15-12): 3042-3045.
[56] Xu D L, Li Z. Controlled projective synchronization in nonpartially-linear chaotic systems. International Journal of Bifurcation and chaos. 2002, 12(6): 1395-1402.
[57] Kim C M, Rim S H, Key W. Anti-synchronization of chaotic oscillators. Physics Letters A. 2003, 320(1): 39-46.
[58] Hu J, Chen S H, Chen L. Adaptive control for anti-synchronization of Chua's chaotic system. Physics Letters A. 2005,339(6): 455-460.
[59] Wen G L, Xu D L. Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems. Chaos, Solitons Fractals. 2005,26(1): 71-77.
[60] Paraskevopoulos P N. Modern control engineering. New York: Marcel Dekker, 2002.