A New Fractional Order Chaotic System with 4-scroll Chaotic Attractors and Its Adaptive Control
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- 英文论文题名：A New Fractional Order Chaotic System with 4-scroll Chaotic Attractors and Its Adaptive Control
- 论文作者：Qiong Wang ; Lingjun Meng ; Keyong Shao ; Xing Han ; Hai Huang ; Kun Qian
- 英文论文作者：Qiong Wang ; Lingjun Meng ; Keyong Shao ; Xing Han ; Hai Huang ; Kun Qian ; School of Electrical and Information Engineering Northeast Petroleum University ; School of Information Technology Bayi Agricultural University ; Sewage Treatment Brigade of Shuguang Oil Extraction Factory China National Petroleum Corporation ; No.3 Oil Mine of The Third Oil Production Plant China National Petroleum Corporation
- 年：2017
- 作者机构：School of Electrical and Information Engineering Northeast Petroleum University;School of Information Technology Bayi Agricultural University;Sewage Treatment Brigade of Shuguang Oil Extraction Factory China National Petroleum Corporation;No.3 Oil Mine of The Third Oil Production Plant China National Petroleum Corporation;
- 英文论文关键词：Fractional order ; Chaotic system ; Chaotic Attractors ; MDTM method ; '0-1 test' ; Lyapunov exponents ; nonlinear controller
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（A）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（A）
- 语种：英文
- 分类号：O231;O415.5
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：7
- 文件大小：631k
- 原文格式：D
- 会议级别：全国
摘要
This paper investigates the coupling of three nonlinear fractional differential equations, which analyzes the dynamic characteristics of the system and uses the modified DTM method to solve the nonlinear fractional differential equation of it. We use the method of '0-1 test' to verify chaotic of the system, compute Lyapunov exponents to verify the regimes of chaotic dynamics, and design a nonlinear controller to influence the trajectory of the system.
This paper investigates the coupling of three nonlinear fractional differential equations, which analyzes the dynamic characteristics of the system and uses the modified DTM method to solve the nonlinear fractional differential equation of it. We use the method of '0-1 test' to verify chaotic of the system, compute Lyapunov exponents to verify the regimes of chaotic dynamics, and design a nonlinear controller to influence the trajectory of the system.
This paper investigates the coupling of three nonlinear fractional differential equations, which analyzes the dynamic characteristics of the system and uses the modified DTM method to solve the nonlinear fractional differential equation of it. We use the method of '0-1 test' to verify chaotic of the system, compute Lyapunov exponents to verify the regimes of chaotic dynamics, and design a nonlinear controller to influence the trajectory of the system.
引文
[1]Eason,G.,B.Noble,and I.N.Sneddon."On Certain Integrals of Lipschitz-Hankel Type Involving Products of Bessel Functions."Philosophical Transactions of the Royal Society A Mathematical Physical&Engineering Sciences 247.935(1955):529-551.
[2]Maxwell,James Clerk.A treatise on electricity and magnetism/.Clarendon,1904.
[3]Jacobs,I.S.,and C.P.Bean."Fine particles,thin films and exchange anisotropy."in Magnetism,vol.III,G.T.Rado and H(1963).
[4]K.Elissa,“Title of paper if known,”unpublished.
[5]R.Nicole,“Title of paper with only first word capitalized,”J.Name Stand.Abbrev,i press.
[6]Li,Chunbiao,et al."A novel four-wing strange attractor born in bistablity."Ieice Electronics Express12.4(2015):20141116-20141116.
[7]V.Sundarapandian,I.Pehliva N,Analysis,control,synchronization,and cir-cuit design of a novel chaotic system,Math.Comput.Model.55(7-8)(2012)1904–1915.
[8]I.Pehlivan,Y.Uyarolu,A new 3D chaotic system with golden proportion equilib-ria:analysis and electronic circuit realization,Comput.Electr.Eng.38(6)(2012)1777–1784.
[9]S.G.Quan,C.Hui,Z.Y.Bin."A new four-dimensional hyperchaotic Lorenz system and its adaptive control."Chinese Physics B 20.1(2011):229-237.
[10]I.Pehlivan,I.M.Moroz,S.Vaidyanathan,Analysis,synchronization and cir-cuit design of a novel butterfly attractor,J.Sound Vib.333(20)(2014)5077–5096
[11]Pehlivan,I."Four-scroll stellate new chaotic system."5.9(2011):1003-1006.
[12]Pehlivan,Ihsan,and Y.Uyaroglu."A new chaotic attractor from general Lorenz system family and its electronic experimental implementation."Turkish Journal of Electrical Engineering&Computer Sciences18.2(2010):171-184.
[13]Jinhu Lü,Guanrong Chen,and Suochun Zhang."Dynamical analysis of a new chaotic attractor."International Journal of Bifurcation&Chaos12.5(2002):1001-1015.
[14]Wang,Zenghui,et al."A hyperchaotic system without equilibrium."Nonlinear Dynamics 69.1-2(2012):531-537.
[15]Liu,Fawang,et al.Fractional differential equations:an introduction to fractional derivatives,fractional differential equations,to methods of their solution and some of their applications.Academic press,1999.
[16]Dalir,Mehdi,and M.Bashour."Applications of fractional calculus."Appl.math.sci 4.21(2010):1021-1032.
[17]Khanzadeh,Alireza,and M.Pourgholi."A novel continuous time-varying sliding mode controller for robustly synchronizing non-identical fractional-order chaotic systems precisely at any arbitrary pre-specified time."Nonlinear Dynamics 86.1(2016):1-16.
[18]Mahmoud,Gamal M.,M.E.Ahmed,and T.M.AbedElhameed."Active control technique of fractional-order chaotic complex systems."The European Physical Journal Plus 131.6(2016):1-11.
[19]Kuntanapreeda,Suwat."Adaptive control of fractionalorder unified chaotic systems using a passivity-based control approach."Nonlinear Dynamics 84.4(2016):2505-2515.
[20]Wang,Xingyuan,and Y.He."Projective synchronization of fractional order chaotic system based on linear separation."Acta Physica Sinica 372.4(2007):435-441.
[21]Pan,Lin,et al."Chaos synchronization between two different fractional-order hyperchaotic systems."Communications in Nonlinear Science&Numerical Simulation 16.6(2011):2628-2640.
[22]Bhalekar,Sachin,and V.Daftardar-Gejji."Synchronization of different fractional order chaotic systems using active control."Communications in Nonlinear Science&Numerical Simulation15.11(2013):3536-3546.
[23]Wang J,Zhang Y.Designing synchronization schemes for chaotic fractional-order unified systems.Chaos,Solitons&Fractals,2006,30(5):1265-1272.
[24]Bataineh,A.S.,et al."Series Solutions of Systems of Nonlinear Fractional Differential Equations."Acta Applicandae Mathematicae 105.2(2009):189-198.
[25]Haeri,Mohammad,and A.A.Emadzadeh."Synchronizing different chaotic systems using active sliding mode control."Chaos Solitons&Fractals 31.1(2007):119-129.
[26]Li,Changpin,and G.Peng."Chaos in Chen's system with a fractional order."Chaos Solitons&Fractals22.2(2004):443-450.
[27]Wang,Junwei,X.Xiong,and Y.Zhang."Extending synchronization scheme to chaotic fractional-order Chen systems."Physica A Statistical Mechanics&Its Applications 370.2(2006):279-285.
[28]Kai,Diethelm."The Analysis of Fractional Differential Equations."Journal of Mathematical Analysis&Applications 265.2(2010):229-248.
[29]Kai,Diethelm,N.J.Ford,and A.D.Freed."A PredictorCorrector Approach for the Numerical Solution of Fractional Differential Equations."Nonlinear Dynamics29.1-4(2002):3-22.
[30]Bhalekar,Sachin."Dynamical analysis of fractional order U?ar prototype delayed system."Signal,Image and Video Processing 6.3(2012):513.
[31]Changpin Li,and Weihua Deng."Chaos Synchronization of fractional-order differential system."Physica D Nonlinear Phenomena 212.1–2(2005):111-125.
[32]Zhou JK.Differential transformation and its applications for electrical circuits(in Chinese).Wuhan,China:Huazhong University Press;1986.
[33]Arikoglu,Aytac,and I.Ozkol."Solution of fractional differential equations by using differential transform method."Chaos Solitons&Fractals 34.5(2007):1473-1481.
[34]Odibat,Z.,S.Momani,and V.S.Erturk."Generalized differential transform method:Application to differential equations of fractional order."Applied Mathematics&Computation 197.197(2008):467-477.
[35]Keskin,Yildiray,and G.Oturan?."Reduced Differential Transform Method for Partial Differential Equations."International Journal of Nonlinear Sciences&Numerical Simulation 10.6(2009):741-750.
[36]Y.Keskin,G.Oturanc,The reduced differential transform method:a new approach to fractional partial differential equations,Nonlinear Sci.Lett.A
[37]Elabbasy,et al."Adaptive synchronization of Lüsystem with uncertain parameters."Chaos Solitons&Fractals21.21(2004):657-667.
[38]Odibat,Zaid M,et al."A multi-step differential transform method and application to non-chaotic or chaotic systems."Computers&Mathematics with Applications59.4(2010):1462-1472.
[39]Alomari,A.K."A new analytic solution for fractional chaotic dynamical systems using the differential transform method."Computers&Mathematics with Applications61.61(2011):2528-2534.
[40]Alomari,A.K."A new analytic solution for fractional chaotic dynamical systems using the differential transform method."Computers&Mathematics with Applications61.61(2011):2528-2534.
[2]Maxwell,James Clerk.A treatise on electricity and magnetism/.Clarendon,1904.
[3]Jacobs,I.S.,and C.P.Bean."Fine particles,thin films and exchange anisotropy."in Magnetism,vol.III,G.T.Rado and H(1963).
[4]K.Elissa,“Title of paper if known,”unpublished.
[5]R.Nicole,“Title of paper with only first word capitalized,”J.Name Stand.Abbrev,i press.
[6]Li,Chunbiao,et al."A novel four-wing strange attractor born in bistablity."Ieice Electronics Express12.4(2015):20141116-20141116.
[7]V.Sundarapandian,I.Pehliva N,Analysis,control,synchronization,and cir-cuit design of a novel chaotic system,Math.Comput.Model.55(7-8)(2012)1904–1915.
[8]I.Pehlivan,Y.Uyarolu,A new 3D chaotic system with golden proportion equilib-ria:analysis and electronic circuit realization,Comput.Electr.Eng.38(6)(2012)1777–1784.
[9]S.G.Quan,C.Hui,Z.Y.Bin."A new four-dimensional hyperchaotic Lorenz system and its adaptive control."Chinese Physics B 20.1(2011):229-237.
[10]I.Pehlivan,I.M.Moroz,S.Vaidyanathan,Analysis,synchronization and cir-cuit design of a novel butterfly attractor,J.Sound Vib.333(20)(2014)5077–5096
[11]Pehlivan,I."Four-scroll stellate new chaotic system."5.9(2011):1003-1006.
[12]Pehlivan,Ihsan,and Y.Uyaroglu."A new chaotic attractor from general Lorenz system family and its electronic experimental implementation."Turkish Journal of Electrical Engineering&Computer Sciences18.2(2010):171-184.
[13]Jinhu Lü,Guanrong Chen,and Suochun Zhang."Dynamical analysis of a new chaotic attractor."International Journal of Bifurcation&Chaos12.5(2002):1001-1015.
[14]Wang,Zenghui,et al."A hyperchaotic system without equilibrium."Nonlinear Dynamics 69.1-2(2012):531-537.
[15]Liu,Fawang,et al.Fractional differential equations:an introduction to fractional derivatives,fractional differential equations,to methods of their solution and some of their applications.Academic press,1999.
[16]Dalir,Mehdi,and M.Bashour."Applications of fractional calculus."Appl.math.sci 4.21(2010):1021-1032.
[17]Khanzadeh,Alireza,and M.Pourgholi."A novel continuous time-varying sliding mode controller for robustly synchronizing non-identical fractional-order chaotic systems precisely at any arbitrary pre-specified time."Nonlinear Dynamics 86.1(2016):1-16.
[18]Mahmoud,Gamal M.,M.E.Ahmed,and T.M.AbedElhameed."Active control technique of fractional-order chaotic complex systems."The European Physical Journal Plus 131.6(2016):1-11.
[19]Kuntanapreeda,Suwat."Adaptive control of fractionalorder unified chaotic systems using a passivity-based control approach."Nonlinear Dynamics 84.4(2016):2505-2515.
[20]Wang,Xingyuan,and Y.He."Projective synchronization of fractional order chaotic system based on linear separation."Acta Physica Sinica 372.4(2007):435-441.
[21]Pan,Lin,et al."Chaos synchronization between two different fractional-order hyperchaotic systems."Communications in Nonlinear Science&Numerical Simulation 16.6(2011):2628-2640.
[22]Bhalekar,Sachin,and V.Daftardar-Gejji."Synchronization of different fractional order chaotic systems using active control."Communications in Nonlinear Science&Numerical Simulation15.11(2013):3536-3546.
[23]Wang J,Zhang Y.Designing synchronization schemes for chaotic fractional-order unified systems.Chaos,Solitons&Fractals,2006,30(5):1265-1272.
[24]Bataineh,A.S.,et al."Series Solutions of Systems of Nonlinear Fractional Differential Equations."Acta Applicandae Mathematicae 105.2(2009):189-198.
[25]Haeri,Mohammad,and A.A.Emadzadeh."Synchronizing different chaotic systems using active sliding mode control."Chaos Solitons&Fractals 31.1(2007):119-129.
[26]Li,Changpin,and G.Peng."Chaos in Chen's system with a fractional order."Chaos Solitons&Fractals22.2(2004):443-450.
[27]Wang,Junwei,X.Xiong,and Y.Zhang."Extending synchronization scheme to chaotic fractional-order Chen systems."Physica A Statistical Mechanics&Its Applications 370.2(2006):279-285.
[28]Kai,Diethelm."The Analysis of Fractional Differential Equations."Journal of Mathematical Analysis&Applications 265.2(2010):229-248.
[29]Kai,Diethelm,N.J.Ford,and A.D.Freed."A PredictorCorrector Approach for the Numerical Solution of Fractional Differential Equations."Nonlinear Dynamics29.1-4(2002):3-22.
[30]Bhalekar,Sachin."Dynamical analysis of fractional order U?ar prototype delayed system."Signal,Image and Video Processing 6.3(2012):513.
[31]Changpin Li,and Weihua Deng."Chaos Synchronization of fractional-order differential system."Physica D Nonlinear Phenomena 212.1–2(2005):111-125.
[32]Zhou JK.Differential transformation and its applications for electrical circuits(in Chinese).Wuhan,China:Huazhong University Press;1986.
[33]Arikoglu,Aytac,and I.Ozkol."Solution of fractional differential equations by using differential transform method."Chaos Solitons&Fractals 34.5(2007):1473-1481.
[34]Odibat,Z.,S.Momani,and V.S.Erturk."Generalized differential transform method:Application to differential equations of fractional order."Applied Mathematics&Computation 197.197(2008):467-477.
[35]Keskin,Yildiray,and G.Oturan?."Reduced Differential Transform Method for Partial Differential Equations."International Journal of Nonlinear Sciences&Numerical Simulation 10.6(2009):741-750.
[36]Y.Keskin,G.Oturanc,The reduced differential transform method:a new approach to fractional partial differential equations,Nonlinear Sci.Lett.A
[37]Elabbasy,et al."Adaptive synchronization of Lüsystem with uncertain parameters."Chaos Solitons&Fractals21.21(2004):657-667.
[38]Odibat,Zaid M,et al."A multi-step differential transform method and application to non-chaotic or chaotic systems."Computers&Mathematics with Applications59.4(2010):1462-1472.
[39]Alomari,A.K."A new analytic solution for fractional chaotic dynamical systems using the differential transform method."Computers&Mathematics with Applications61.61(2011):2528-2534.
[40]Alomari,A.K."A new analytic solution for fractional chaotic dynamical systems using the differential transform method."Computers&Mathematics with Applications61.61(2011):2528-2534.
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