一种Logistic分数阶时滞方程混沌特性的研究
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摘要
提出一种新的Logistic分数阶时滞方程,利用预估-校正算法,研究该方程的数值解。并采用高可靠性的0-1混沌检验方法来判断该分数阶时滞方程是否存在混沌行为。仿真结果表明关键参数在有效取值范围时,该Logistic分数阶时滞方程呈现混沌动力学行为。实验结果证明了所采用方法的有效性,同时进一步验证理论分析与实验结果的一致性。
引文
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[2]黄丽莲,周晓亮,项建弘.分数阶PID控制器参数的自适应设计[J].系统工程与电子技术,2013,35(5):1064-1069.
[3]徐争辉,刘友金,谭文,等.一个对称分数阶经济系统混沌特性研究[J].系统工程理论与实践,2014,34(5):1237-1242.
[4]Liguo Yuan,Qigui Yang,Caibin Zeng.Chaos detection and parameter identification in fractional-order chaotic systems with delay[J].Nonlinear Dynamics,2013,73(1-2):439-448
[5]Petrá?,I.Fractional-Order Nonlinear Systems:Modeling,Analysis and Simulation[M].Higher Education Press,Beijing(2011).
[6]Debnath,L.Recent applications of fractional calculus to science and engineering[J].International Journal of Applied Mathematics and Computers Science.54,3413-3442(2003).
[7]Bhalekar,S.,Daftardar-Gejji,V.A predictor–corrector scheme for solving nonlinear delay differential equations of fractional order[J].Journal of Fractional Calculus and Applied Analysis,5,1-9(2011).
[8]Daftardar-Gejji,V.,Bhalekar,S.,Gade,P.:Dynamics of fractional-ordered Chen system with delay.Pram-ana 79,61-69(2012).
[9]Tang,J.E.,Zou,C.R.,Zhao,L.:Chaos synchronization of fractional order time-delay Chen system and its application in secure communication.J.Converg.Inf.Technol.7,124-131(2012).
[10]Wang,Z.,Huang,X.,Shi,G.D.:Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay.Comput.Math.Appl.62,1531-1539(2011).
[11]Rosenstein MT,Collins JJ,Luca CJD.A practical method for calculating largest lyapunov exponents from small data sets.Physica D 1993;65:117-34.
[12]de la Fraga LG,Tlelo-Cuautle E.Optimizing the maximum lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators.Nonlinear Dyn2014;76:1503-15.
[13]Kilbas,A.A.,Srivastava,H.M.,Trujillo,J.J.Theory and Application of Fractional Differential Equations[M].Elsevier,Amsterdam(2006).
[14]Petrá?,I.Fractional-Order Nonlinear Systems:Modeling,Analysis and Simulation[M].Higher Education Press,Beijing(2011).
[15]Diethelm,K.,Ford,N.,Freed,A.A predictor–corrector approach for the numerical solution of fractional differential equations[J].Nonlinear Dyn.29,3-22(2002).
[16]Bhalekar,S.,Daftardar-Gejji,V.A predictor–corrector scheme for solving nonlinear delay differential equations of fractional order[J].J.Fract.Calc.Appl.5,1-9(2011).
[17]Gottwald,G.A.,Melbourne,I.A new test for chaos in deterministic systems[J].Proc.R.Soc.Lond.Ser.A,Math.Phys.Sci.460,603-611(2004).
[18]Gottwald,G.A.,Melbourne,I.On the implementation of the 0–1 test for chaos[J].SIAM J.Appl.Dyn.Syst.8,129-145(2009).
[19]Gottwald,G.A.,Melbourne,I.On the validity of the 0–1 test for chaos[J].Nonlinearity 22,1367-1382(2009).