二维Logistic分数阶微分方程的离散化过程
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摘要
针对二维Logistic分数阶微分方程的求解问题,引进了一种离散化方法对其进行离散求解。首先,将二维Logistic整数阶微分方程推广到分数阶微积分领域;其次,分析相应具有分段常数变元的二维Logistic分数阶微分方程并应用提出的离散化方法对模型进行数值求解;然后,根据不动点理论讨论该合成动力系统不动点的稳定性,给出了在参数空间内二维Logistic分数阶系统发生第一次分岔的边界方程;最后,借助Matlab对模型进行数值仿真,并结合Lyapunov指数、相图、时间序列图、分岔图探讨模型更多复杂的动力学现象。仿真结果显示,所提方法成功对二维Logistic分数阶微分方程进行离散。
Focusing on the problem of solving coupled Logistic fractional-order differential equation, a discretization method was introduced to solve it discretly. Firstly, a coupled Logistic integer-order differential equation was introduced into the fields of fractional-order calculus. Secondly, the corresponding coupled Logistic fractional-order differential equation with piecewise constant arguments was analyzed and the proposed discretization method was applied to solve the model numerically.Then, according to the fixed point theory, the stability of the fixed point of the synthetic dynamic system was discussed, and the boundary equation of the first bifurcation of the coupled Logistic fractional-order system in the parameter space was given.Finally, the model was numerically simulated by Matlab, and more complex dynamics phenomena of model were discussed with Lyapunov index, phase diagram, time series diagram and bifurcation diagram. The simulation results show that, the proposed method is successful in discretizing coupled Logistic fractional-order differential equation.
Focusing on the problem of solving coupled Logistic fractional-order differential equation, a discretization method was introduced to solve it discretly. Firstly, a coupled Logistic integer-order differential equation was introduced into the fields of fractional-order calculus. Secondly, the corresponding coupled Logistic fractional-order differential equation with piecewise constant arguments was analyzed and the proposed discretization method was applied to solve the model numerically.Then, according to the fixed point theory, the stability of the fixed point of the synthetic dynamic system was discussed, and the boundary equation of the first bifurcation of the coupled Logistic fractional-order system in the parameter space was given.Finally, the model was numerically simulated by Matlab, and more complex dynamics phenomena of model were discussed with Lyapunov index, phase diagram, time series diagram and bifurcation diagram. The simulation results show that, the proposed method is successful in discretizing coupled Logistic fractional-order differential equation.
引文
[1]TONG X J. Design of an image encryption scheme based on a multiple chaotic map[J]. Communications in Nonlinear Science and Numerical Simulation, 2013, 18(7):1725-1733.
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[3]LEE T F. Enhancing the security of password authenticated key agreement protocols based on chaotic maps[J]. Information Sciences, 2015, 290:63-71.
[4]刘泉,李佩玥,章明朝,等.基于可Markov分割混沌系统的图像加密算法[J].电子与信息学报,2014,36(6):1271-1277.(LIU Q, LI P Y, ZHANG M C, et al. Image encryption algorithm based on chaos system having Markov portion[J]. Journal of Electronics&Information Technology, 2014, 36(6):1271-1277.)
[5]MAY R M. Simple mathematical models with very complicated dynamics[J]. Nature, 1976, 261(5560):459-467.
[6]FEIGENBAUM M J. Quantitative universality for a class of nonlinear transformations[J]. Journal of Statistical Physics, 1978, 19(1):25-52.
[7]RANI M, KUMAR V. A new experiment with the logistic function[J]. Journal of the Indian Academy of Mathematics, 2005, 27(12):143-156.
[8]VZQUEZ-MEDINA R, DAZ-MNDEZ A, del RO-CORREA J L, et al. Design of chaotic analog noise generators with logistic mapand MOS QT circuits[J]. Chaos, Solitons&Fractals, 2009, 40(4):1779-1793.
[9]RANI M, AGARWAL R. A new experimental approach to study the stability of logistic map[J]. Chaos, Solitons&Fractals, 2009, 41(4):2062-2066.
[10]SAKAGUCHI H, TOMITA K. Bifurcations of the coupled logistic map[J]. Progress of Theoretical Physics, 1987, 78(2):305-315.
[11]WELSTEAD S T, CROMER T L. Coloring periodicities of two-dimensional mappings[J]. Computers&Graphics, 1989, 13(4):539-543.
[12]王兴元,王明军.二维Logistic映射的混沌控制[J].物理学报,2008,57(2):731-736.(WANG X Y, WANG M J. Chaos control of the coupled Logistic map[J]. Acta Physica Sinica, 2008, 57(2):731-736.)
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[14]王兴元,朱伟勇.二维Logistic映射中混沌与分形的研究[J].中国图象图形学报,1999,4A(4):340-344.(WANG X Y, ZHU W Y. Researches on chaos and fractal of the coupled Logistic map[J]. Journal of Image and Graphics, 1999, 4A(4):340-344.)
[15]王兴元,梁庆永.复合Logistic映射中的逆分岔与分形[J].力学学报,2005,37(4):522-528.(WANG X Y, LIANG Q Y. Reverse bifurcation and fractal of a compound Logistic map[J]. Chinese Journal of Theoretical and Applied Mechanics, 2005, 37(4):522-528.)
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[18]HARTLEY T T, LORENZO C F, KILLORY QAMMER H. Chaos in a fractional order Chua's system[J]. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 1995,42(8):485-490.
[19]MICHAEL R B. Strange Attractors:Chaos, Complexity, and the Art of Family Therapy[M]. New York:John Wiley&Sons Inc,1996:3-76.
[20]程金发.分数阶差分方程理论[M].厦门:厦门大学出版社,2011:30-43.(CHENG J F. Theory of Fractional Differential Equations[M]. Xiamen:Xiamen University Press, 2011:30-43.)
[21]SUN H, ABDELWAHED A, ONARAL B. Linear approximation for transfer function with a pole of fractional-order[J]. IEEE Transactions on Automatic Control, 1984, 29(5):441-444.
[22]DIETHELM K, FORD N J, FREEd A D. A predictor-corrector approach for the numerical solution of fractional differential equations[J]. Nonlinear Dynamics, 2002, 29(1/2/3/4):3-22.
[23]SHUKLA M K, SHARMA B B. Stabilization of fractional order discrete chaotic systems[M]//AZAR A, VAIDYANATHAN S,OUANNAS A. Fractional Order Control and Synchronization of Chaotic Systems. Berlin:Springer, 2017:431-445.
[24]陈斯养,张艳.具有分段常数变量的捕食-被捕食模型的分支分析[J].兰州大学学报(自然科学版),2012,48(3):103-112,117.(CHEN S Y, ZHANG Y. Stability and bifurcation analysis of a predator-prey model with piecewise constant arguments[J]. Journal of Lanzhou University(Natural Sciences), 2012, 48(3):103-112, 117.)
[25]林诗仲,俞元洪.含分段常数变元的中立型时滞微分方程的振动定理[J].数学杂志,1997,17(1):143-144.(LIN S Z, YU Y H. The oscillation theory of neutral delay differential equations with piecewise constant arguments[J]. Journal of Mathematics, 1997,17(1):143-144.)
[26]陈斯养,朱晓琳.具有时滞和分段常数变量的单种群收获模型的分支分析[J].陕西师范大学学报(自然科学版),2013,41(2):1-4.(CHEN S Y, ZHU X L. Bifurcation analysis of the single population harvest model with time-delay and piecewise constant variables[J]. Journal of Shaanxi Normal University(Natural Science Edition), 2013, 41(2):1-4.)
[27]EL-RAHEEM Z F, SALMAN S M. On a discretization process of fractional-order Logistic differential equation[J]. Journal of the Egyptian Mathematical Society, 2014, 22(3):407-412.
[28]El-SAYED A M A, SALMAN S M. Chaos and bifurcation of the logistic discontinuous dynamical systems with piecewise constant arguments[J]. Malaya Journal of Matematik, 2013, 3(1):14-20.
[29]El-SAYED A M A, SALMAN S M. On a discretization process of fractional order riccati differential equation[J]. Journal of Fractional Calculus and Application, 2013, 4(2):251-259.
[30]El-SAYED A M A, SALMAN S M, ELABD N A. On a fractionalorder delay Mackey-Glass equation[J]. Advances in Difference Equations, 2016, 2016:137-1-137-11.
[31]王兴元,骆超.二维Logistic映射的动力学分析[J].软件学报,2006,17(4):729-739.(WANG X Y, LUO C. Dynamic analysis of the coupled Logistic map[J]. Journal of Software, 2006, 17(4):729-739.)
[32]王兴元.复杂非线性系统中的混沌[M].北京:电子工业出版社,2003:91-113.(WANG X Y. Chaos in Complex Nonlinear Systems[M]. Beijing:Publishing House of Electronics Industry,2003:91-113.)
[33]ECKMANN J-P. Roads to turbulence in dissipative dynamics system[J]. Review Modern Physics, 1981, 53(4):643-654.
[34]古元风,肖剑. Willis环脑动脉瘤系统的混沌分析及随机相位控制[J].物理学报,2004,63(16):51-58.(GU Y F, XIAO J.Analysis of the Willis chaotic system and the control of random phase[J]. Acta Physical Sinica, 2004, 63(16):51-58.)
[2]陈志刚,梁涤青,邓小鸿,等. Logistic混沌映射性能分析与改进[J].电子与信息学报,2016,38(6):1547-1551.(CHEN Z G,LIANG D Q, DENG X H, et al. Performance analysis and improvement of Logistic chaotic mapping[J]. Journal of Electronics&Information Technology, 2016, 38(6):1547-1551.)
[3]LEE T F. Enhancing the security of password authenticated key agreement protocols based on chaotic maps[J]. Information Sciences, 2015, 290:63-71.
[4]刘泉,李佩玥,章明朝,等.基于可Markov分割混沌系统的图像加密算法[J].电子与信息学报,2014,36(6):1271-1277.(LIU Q, LI P Y, ZHANG M C, et al. Image encryption algorithm based on chaos system having Markov portion[J]. Journal of Electronics&Information Technology, 2014, 36(6):1271-1277.)
[5]MAY R M. Simple mathematical models with very complicated dynamics[J]. Nature, 1976, 261(5560):459-467.
[6]FEIGENBAUM M J. Quantitative universality for a class of nonlinear transformations[J]. Journal of Statistical Physics, 1978, 19(1):25-52.
[7]RANI M, KUMAR V. A new experiment with the logistic function[J]. Journal of the Indian Academy of Mathematics, 2005, 27(12):143-156.
[8]VZQUEZ-MEDINA R, DAZ-MNDEZ A, del RO-CORREA J L, et al. Design of chaotic analog noise generators with logistic mapand MOS QT circuits[J]. Chaos, Solitons&Fractals, 2009, 40(4):1779-1793.
[9]RANI M, AGARWAL R. A new experimental approach to study the stability of logistic map[J]. Chaos, Solitons&Fractals, 2009, 41(4):2062-2066.
[10]SAKAGUCHI H, TOMITA K. Bifurcations of the coupled logistic map[J]. Progress of Theoretical Physics, 1987, 78(2):305-315.
[11]WELSTEAD S T, CROMER T L. Coloring periodicities of two-dimensional mappings[J]. Computers&Graphics, 1989, 13(4):539-543.
[12]王兴元,王明军.二维Logistic映射的混沌控制[J].物理学报,2008,57(2):731-736.(WANG X Y, WANG M J. Chaos control of the coupled Logistic map[J]. Acta Physica Sinica, 2008, 57(2):731-736.)
[13]张硕,蔡如华,陈光喜.结合二维混沌映射与小波变换的图像加密方案[J].计算机工程与应用,2010,46(33):191-194.(ZHANG S, CAI R H, CHEN G X. Image encryption algorithm on coupled chaotic map and wavelet transform[J]. Computer Engineering and Applications, 2010, 46(33):191-194.)
[14]王兴元,朱伟勇.二维Logistic映射中混沌与分形的研究[J].中国图象图形学报,1999,4A(4):340-344.(WANG X Y, ZHU W Y. Researches on chaos and fractal of the coupled Logistic map[J]. Journal of Image and Graphics, 1999, 4A(4):340-344.)
[15]王兴元,梁庆永.复合Logistic映射中的逆分岔与分形[J].力学学报,2005,37(4):522-528.(WANG X Y, LIANG Q Y. Reverse bifurcation and fractal of a compound Logistic map[J]. Chinese Journal of Theoretical and Applied Mechanics, 2005, 37(4):522-528.)
[16]王兴元,朱伟勇,顾树生.一般二维二次映射中的混沌与分形[J].计算机辅助设计与图形学学报,2000,12(6):408-413.(WANG X Y, ZHU W Y, GU S S. Chaos and fractal of the general two-dimensional quadratic map[J]. Journal of Computer Aided Design and Computer Graphics, 2000, 12(6):408-413.)
[17]朱伟勇,王兴元.二维Logistic映射中的阵发混沌与分形[J].东北大学学报(自然科学版),1998,19(5):509-512.(ZHU W Y,WANG X Y. Intermittent chaos and fractal of the coupled Logistic map[J]. Journal of Northeastern University(Natural Science),1998, 19(5):509-512.)
[18]HARTLEY T T, LORENZO C F, KILLORY QAMMER H. Chaos in a fractional order Chua's system[J]. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 1995,42(8):485-490.
[19]MICHAEL R B. Strange Attractors:Chaos, Complexity, and the Art of Family Therapy[M]. New York:John Wiley&Sons Inc,1996:3-76.
[20]程金发.分数阶差分方程理论[M].厦门:厦门大学出版社,2011:30-43.(CHENG J F. Theory of Fractional Differential Equations[M]. Xiamen:Xiamen University Press, 2011:30-43.)
[21]SUN H, ABDELWAHED A, ONARAL B. Linear approximation for transfer function with a pole of fractional-order[J]. IEEE Transactions on Automatic Control, 1984, 29(5):441-444.
[22]DIETHELM K, FORD N J, FREEd A D. A predictor-corrector approach for the numerical solution of fractional differential equations[J]. Nonlinear Dynamics, 2002, 29(1/2/3/4):3-22.
[23]SHUKLA M K, SHARMA B B. Stabilization of fractional order discrete chaotic systems[M]//AZAR A, VAIDYANATHAN S,OUANNAS A. Fractional Order Control and Synchronization of Chaotic Systems. Berlin:Springer, 2017:431-445.
[24]陈斯养,张艳.具有分段常数变量的捕食-被捕食模型的分支分析[J].兰州大学学报(自然科学版),2012,48(3):103-112,117.(CHEN S Y, ZHANG Y. Stability and bifurcation analysis of a predator-prey model with piecewise constant arguments[J]. Journal of Lanzhou University(Natural Sciences), 2012, 48(3):103-112, 117.)
[25]林诗仲,俞元洪.含分段常数变元的中立型时滞微分方程的振动定理[J].数学杂志,1997,17(1):143-144.(LIN S Z, YU Y H. The oscillation theory of neutral delay differential equations with piecewise constant arguments[J]. Journal of Mathematics, 1997,17(1):143-144.)
[26]陈斯养,朱晓琳.具有时滞和分段常数变量的单种群收获模型的分支分析[J].陕西师范大学学报(自然科学版),2013,41(2):1-4.(CHEN S Y, ZHU X L. Bifurcation analysis of the single population harvest model with time-delay and piecewise constant variables[J]. Journal of Shaanxi Normal University(Natural Science Edition), 2013, 41(2):1-4.)
[27]EL-RAHEEM Z F, SALMAN S M. On a discretization process of fractional-order Logistic differential equation[J]. Journal of the Egyptian Mathematical Society, 2014, 22(3):407-412.
[28]El-SAYED A M A, SALMAN S M. Chaos and bifurcation of the logistic discontinuous dynamical systems with piecewise constant arguments[J]. Malaya Journal of Matematik, 2013, 3(1):14-20.
[29]El-SAYED A M A, SALMAN S M. On a discretization process of fractional order riccati differential equation[J]. Journal of Fractional Calculus and Application, 2013, 4(2):251-259.
[30]El-SAYED A M A, SALMAN S M, ELABD N A. On a fractionalorder delay Mackey-Glass equation[J]. Advances in Difference Equations, 2016, 2016:137-1-137-11.
[31]王兴元,骆超.二维Logistic映射的动力学分析[J].软件学报,2006,17(4):729-739.(WANG X Y, LUO C. Dynamic analysis of the coupled Logistic map[J]. Journal of Software, 2006, 17(4):729-739.)
[32]王兴元.复杂非线性系统中的混沌[M].北京:电子工业出版社,2003:91-113.(WANG X Y. Chaos in Complex Nonlinear Systems[M]. Beijing:Publishing House of Electronics Industry,2003:91-113.)
[33]ECKMANN J-P. Roads to turbulence in dissipative dynamics system[J]. Review Modern Physics, 1981, 53(4):643-654.
[34]古元风,肖剑. Willis环脑动脉瘤系统的混沌分析及随机相位控制[J].物理学报,2004,63(16):51-58.(GU Y F, XIAO J.Analysis of the Willis chaotic system and the control of random phase[J]. Acta Physical Sinica, 2004, 63(16):51-58.)