Abstract
Motivated by the importance of study on the complex behaviors, which may be exhibited by fractional order differential equations, this review paper focuses on dynamical fractional order systems exhibiting chaotic behaviors. The review begins with a brief history on the first publications on the above-mentioned subject. Then, the review is continued by investigating the recent progresses relevant to fractional order chaotic systems. Furthermore, a summary on some applications for such systems, which have been reported in the literature, is presented. Finally, the paper is closed by discussing some open problems on the aforementioned research subject. These open problems, as future challenges for further study on fractional order chaotic systems, can specify some direction lines for continuing the research on that subject.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Ross, Hist. Math. 4, 75 (1977)
B. Ross, Math. Mag. 50, 115 (1977)
D. Cafagna, IEEE Ind. Electron. Mag. 1, 35 (2007)
M.D. Ortigueira, IEEE Circuits Syst. Mag. 8, 19 (2008)
A.S. Elwakil, IEEE Circuits Syst. Mag. 10, 40 (2010)
L. Chen, J. Zhang, J. Zhao, W. Cao, H. Wang, J. Zhang, Comput. Phys. Commun. 245, 106842 (2019)
T. Jin, Y. Sun, Y. Zhu, Physica A 534, 122357 (2019)
R. Brociek, D. Słota, M. Król, G. Matula, W. Kwas′ny, Int. J. Heat Mass Transfer 143, 118440 (2019)
M. Amabili, P. Balasubramanian, I. Breslavsky, J. Mech. Behav. Biomed. Mater. 99, 186 (2019)
R.K. Miller, T.C. Walker, Fractals and Chaos: Exploiting Real-World Applications (SEAI Technical Publications, 1991)
G.L. Baker, J.P. Gollub, Chaotic Dynamics: An Introduction, 2nd ed. (Cambridge University Press, 2012)
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
K. Diethelm, The Analysis of Fractional Differential Equations (Springer, Berlin, 2010)
M. Tavakoli-Kakhki, M. Haeri, M.S. Tavazoei, IEEE Trans. Circuits Syst. I: Regul. Pap. 58, 1099 (2011)
T.T. Hartley, C.F. Lorenzo, H.K. Qammer, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 42, 485 (1995)
H.K. Khalil, Nonlinear Systems. 3rd ed. (Pearson, 2001)
T.T. Hartley, F. Mossayebi, J. Circuits Syst. Comput. 3, 173 (1993)
P. Arena, R. Caponetto, L. Fortuna, D. Porto, Int. J. Bifurc. Chaos 8, 1527 (1998)
S. Nimmo, A.K. Evans, Chaos Solitons Fractals 10, 1111 (1999)
J.C. Sprott, Phys. Rev. E 50, 647 (1994)
I. Grigorenko, E. Grigorenko, Phys. Rev. Lett. 91, 034101 (2003)
I. Grigorenko, E. Grigorenko, Phys. Rev. Lett. 96,199902 (2006)
C. Li, G. Chen, Physica A 341, 55 (2004)
C. Li, G. Chen, Chaos Solitons Fractals 22, 549 (2004)
J.G. Lu, Phys. Lett. A 354, 305 (2006)
J.G. Lu, Chaos Solitons Fractals 26, 1125 (2005)
J.G. Lu, Chin. Phys. 14, 1517 (2005)
J.G. Lu, Chin. Phys. 15, 301 (2006)
G.M. Zaslavsky, A.A. Stanislavsky, M. Edelman, Chaos 16, 013102 (2006)
M. Seredynska, A. Hanyga, Int. J. Bifurc. Chaos 14, 1291 (2004)
F.B.M. Duarte, J.A.T. Machado, Nonlinear Dyn. 22, 315 (2002)
W.M. Ahmad, A.M. Harb, Chaos Solitons Fractals 18, 693 (2003)
I. Petras, Acta Montanistica Slovaca 11, 273 (2006)
W.M. Ahmad, J.C. Sprott, Chaos Solitons Fractals 16, 339 (2003)
M.S. Tavazoei, M. Haeri, Phys. Lett. A 367, 102 (2007)
M.S. Tavazoei, M. Haeri, Physica D 237, 2628 (2008)
M.S. Tavazoei, IEEE Trans. Circuits Syst. II: Express Briefs 56, 519 (2009)
M.S. Tavazoei, IEEE Trans. Circuits Syst. I: Regul. Pap. 62, 329 (2015)
M.S. Tavazoei, M. Haeri, IET Signal Proc. 1, 171 (2007)
M.S. Tavazoei, M. Haeri, S. Bolouki, M. Siami, SIAM J. Numer. Anal. 47, 321 (2008)
M.S. Tavazoei, M. Haeri, Automatica 46, 94 (2010)
X. Lin, B. Liao, C. Zeng, Int. J. Bifurc. Chaos 27, 1750207 (2017)
J. Čermák, L. Nechvátal, Int. J. Bifurc. Chaos 28, 1850098 (2018)
Z. Odibat, N. Corson, M.A. Aziz-Alaoui, A. Alsaedi, Int. J. Bifurc. Chaos 27, 1750161 (2017)
E.F.D. Goufo, Int. J. Bifurc. Chaos 28, 1850125 (2018)
M. Borah, B.K. Roy, Chaos Solitons Fractals 102, 372 (2017)
E. Bonyah, Chaos Solitons Fractals 116, 316 (2018)
E.F.D. Goufo, Chaos Solitons Fractals 104, 443 (2017)
L. Chen, W. Pan, R. Wu, J.A. Tenreiro Machado, A.M. Lopes, Chaos 26, 084303 (2016)
J. Palanivel, K. Suresh, S. Sabarathinam, K. Thamilmaran, Chaos Solitons Fractals 95, 33 (2017)
M.P. Kennedy, IEEE Trans. Circuits Syst. I: Fundam. Theor. Appl. 40, 657 (1993)
D.B. Strukov, G.S. Snider, D.R. Stewart, R.S. Williams, Nature 453, 80 (2008)
I. Petras, IEEE Trans. Circuits Syst. II: Express Briefs 57, 975 (2010)
D. Cafagna, G. Grassi, Nonlinear Dyn. 70, 1185 (2012)
J. Ruan, K. Sun, J. Mou, S. He, L. Zhang, Eur. Phys. J. Plus 133, 3 (2018)
L. Teng, H.H.C. Iu, X. Wang, X. Wang, Nonlinear Dyn. 77, 231 (2014)
W. Hu, D. Ding, Y. Zhang, N. Wang, D. Liang, Optik 130, 189 (2017)
Y. Fan, X. Huang, Z. Wang, Y. Li, Nonlinear Dyn. 93, 611 (2018)
K. Rajagopal, A. Karthikeyan, A. Srinivasan, Nonlinear Dyn. 91, 1491 (2018)
K. Rajagopal, S.T. Kingni, A.J.M. Khalaf, Y. Shekofteh, F. Nazarimehr, Eur. Phys. J. Special Topics 228, 2035 (2019)
A.G. Soriano-Sánchez, C. Posadas-Castillo, M.A. Platas-Garza, A. Arellano-Delgado, Appl. Math. Comput. 332, 250 (2018)
D.K. Shah, R.B. Chaurasiya, V.A. Vyawahare, K. Pichhode, M.D. Patil, Int. J. Electron. Commun. 78, 245 (2017)
X. Zhang, Z. Li, D. Chang, Int. J. Electron. Commun. 82, 435 (2017)
L.F. Ávalos-Ruiz, C.J. Zúñiga-Aguilar, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, H.M. Romero-Ugalde, Chaos Solitons Fractals 115, 177 (2018)
D. Liu, Y. Xu, J. Li, J. Sound Vib. 399, 182 (2017)
P.R. Nwagoum Tuwa, C.H. Miwadinou, A.V. Monwanou, J.B. Chabi Orou, P. Woafo, Mech. Res. Commun. 97, 8 (2019)
L.M. Anague Tabejieu, B.R. Nana Nbendjo, P. Woafo, Chaos Solitons Fractals 93, 39 (2016)
Y. Long, B. Xu, D. Chen, W. Ye, Appl. Math. Model. 58, 128 (2018)
H. Ye, Y. Ding, Math. Prob. Eng. 2009, 378614 (2009)
I. N’Doye, H. Voos, M. Darouach, Proc. Eur. Control Conf. 2014, 171 (2014)
B. Bodo, A. Mvogo, S. Morfu, Chaos Solitons Fractals 102, 426 (2017)
M. Das, A. Maiti, G.P. Samanta, Ecol. Genet. Genomics 7–8, 33 (2018)
C. Wen, J. Yang, Chaos Solitons Fractals 128, 242 (2019)
G. Zhang, P. Qian, Z. Su, Chaos Solitons Fractals 128, 219 (2019)
H. Wang, C. Weng, Z. Song, J. Cai, Chaos Solitons Fractals 131, 109462 (2020)
C. Huang, L. Cai, J. Cao, Chaos Solitons Fractals 113, 326 (2018)
V.T. Pham, S.T. Kingni, C. Volos, S. Jafari, T. Kapitaniak, Int. J. Electron. Commun. 78, 220 (2017)
S. Shao, M. Chen, IEEE/CAA J. Autom. Sinica 6, 1000 (2019)
V.T. Pham, A. Ouannas, C. Volos, T. Kapitaniak, Int. J. Electron. Commun. 86, 69 (2018)
J. Mishra, Chaos Solitons Fractals 116, 43 (2018)
E.F.D. Goufo, Chaos Solitons Fractals 119 (2019) 24.
S. Fang, Z. Li, X. Zhang, Y. Li, Braz. J. Phys. 49, 846 (2019)
K. Rajagopal, A. Karthikeyan, A.K. Srinivasan, Nonlinear Dyn. 87, 2281 (2017)
D. Cafagna, G. Grassi, Chin. Phys. B 24, 080502 (2015)
J.M. Munoz-Pacheco, E. Zambrano-Serrano, C. Volos, S. Jafari, J. Kengne, K. Rajagopal, Entropy 20, 564 (2018)
J.M. Munoz-Pacheco, Eur. Phys. J. Special Topics 228, 2185 (2019)
S.T. Kingni, S. Jafari, H. Simo, P. Woafo, Eur. Phys. J. Plus 129, 76 (2014)
Z. Wei, A. Akgul, U.E. Kocamaz, I. Moroz, W. Zhang, Chaos Solitons Fractals 111, 157 (2018)
C. Volos, V.T. Pham, E. Zambrano-Serrano, J.M. Munoz-Pacheco, S. Vaidyanathan, E. Tlelo-Cuautle, Fundamental Advances in memristors, memristive devices and systems, in Studies in Computational Intelligence, edited by S. Vaidyanathan, C. Volos (Springer, 2017), Vol. 3, pp. 207–235
S.T. Kingni, V.T. Pham, S. Jafari, P. Woafo, Chaos Solitons Fractals 99, 209 (2017)
S.T. Kingni, V.T. Pham, S. Jafari, G.R. Kol, P. Woafo, Circuits Syst Signal Process. 35, 1933 (2016)
M.F. Danca, Nonlinear Dyn. 89, 577 (2017)
C. Goodrich, A.C. Peterson, Discrete Fractional Calculus (Springer, 2015)
G.C. Wu, D. Baleanu, Nonlinear Dyn. 75, 283 (2014)
Y.D. Ji, L. Lai, S.C. Zhong, L. Zhang, Commun. Nonlinear Sci. Numer. Simul. 57, 352 (2018)
G.C. Wu, D. Baleanu, S.D. Zeng, Phys. Lett. A 378, 484 (2014)
A. Ouannas, A.A. Khennaoui, Z. Odibat, V.T. Pham, G. Grassi, Chaos Solitons Fractals 123, 108 (2019)
L. Huang, L. Wang, D. Shi, IEEE/CAA J. Autom. Sinica, https://doi.org/10.1109/JAS.2016.7510148
M.K. Shukla, B.B. Sharma, Int. J. Electron. Commun. 78, 265 (2017)
A.A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, R. Lozi, V.T. Pham, Chaos Solitons Fractals 119, 150 (2019)
A. Ouannas, A.A. Khennaoui, G. Grassi, S. Bendoukha, J. Comput. Appl. Math. 358, 293 (2019)
A.A. Khennaoui, A. Ouannas, S. Bendoukha, X. Wang, V.T. Pham, Entropy 20, 530 (2018)
Y. Peng, K. Sun, D. Peng, W. Ai, Physica A 525, 96 (2019)
S. He, K. Sun, Y. Peng, Phys. Lett. A 383, 2267 (2019)
Z. Liu, T. Xia, Appl. Comput. Inf. 14, 177 (2018)
A. Ouannas, A.A. Khennaoui, S. Bendoukha, G. Grassi, Int. J. Bifurc. Chaos 29, 1950078 (2019)
R. Almeida, D. Tavares, D.F.M. Torres, The Variable-order Fractional Calculus of Variations, Springer Briefs in Applied Sciences and Technology (Springer, 2019)
S. He, S. Banerjee, K. Sun, Chaos Solitons Fractals 115, 14 (2018)
C.J. Zuñiga-Aguilar, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, H.M. Romero-Ugalde, Eur. Phys. J. Plus 133, 13 (2018)
L. Zhang, Ch. Yu, T. Liu, Nonlinear Dyn. 86, 1967 (2016)
L.F. Ávalos-Ruiz, C.J. Zúñiga-Aguilar, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, H.M. Romero-Ugalde, Chaos Solitons Fractals 115, 083103 (2018)
G.C. Wu, Z.G. Deng, D. Baleanu, D.Q. Zeng, Chaos 29, 083103 (2019)
G.C. Wu, D. Baleanu, H.P. Xie, S.D. Zeng, Int. J. Bifurc. Chaos 26, 1650013 (2016)
Z. Jiao, Y.Q. Chen, I. Podlubny, Distributed-order Dynamic Systems: Stability, Simulation, Applications and Perspectives, Springer Briefs in Electrical and Computer Engineering/Springer Briefs in Control, Automation and Robotics (Springer, 2012)
H. Taghavian, M.S. Tavazoei, Int. J. Syst. Sci. 49, 523 (2018)
T.T. Hartley, C.F. Lorenzo, Int. J. Bifurc. Chaos 22, 1250253 (2012)
C. Li, Z. Gong, D. Qian, Y.Q. Chen, Chaos 20, 013127 (2010)
R. Caponetto, S. Fazzino, Commun. Nonlinear Sci. Numer. Simul. 18, 22 (2013)
M.F. Danca, Nonlinear Dyn. 81, 227 (2015)
A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16, 285 (1985)
M.F. Danca, N. Kuznetsov, Int. J. Bifurc. Chaos 28, 1850067 (2018)
K. Diethelm, N. Ford, A. Freed, Nonlinear Dyn. 29, 3 (2002)
B.R. Andrievskii, A.L. Fradkov, Autom. Remote Control 64, 673 (2003)
B.R. Andrievskii, A.L. Fradkov, Autom. Remote Control 65, 505 (2004)
M.S. Tavazoei, J. Comput. Nonlinear Dyn. 6, 031012 (2011)
M.S. Tavazoei, M. Haeri, Eur. J. Control 14, 247 (2008)
M.S. Tavazoei, M. Haeri, J. Dyn. Syst. Meas. Control 132, 021008 (2010)
B. Aguiar, T. González, M. Bernal, IEEE Trans. Autom. Control 61, 2796 (2016)
R. Luo, H. Su, Complexity 2018, 2796 (2018)
S. Zhang, H. Liu, S. Li, Adv. Difference Equ. 2018, 412 (2018)
Y. Zheng, Y. Nian, D. Wang, Phys. Lett. A 375, 125 (2010)
K. Khettab, S. Ladaci, Y. Bensafia, IEEE/CAA J. Autom. Sinica 6, 125 (2019)
L. Shaohua, L. Shaobo, F. Tajaddodianfar, H. Jianjun, J. Franklin Inst. 355, 6435 (2018)
R. Li, W. Li, Optik 126, 2965 (2015)
Y. Zheng, Z. Ji, Chaos Solitons Fractals 87, 307 (2016)
Y. Zheng, Optik 126, 5645 (2015)
Y. Wu, H. Lv, AIP Adv. 6, 085121 (2016)
G. Li, C. Sun, Adv. Difference Equ. 2019, 148 (2019)
B. Bourouba, S. Ladaci, Algorithms 11, 101 (2018)
J.S.H. Tsai, T.H. Chien, S.M. Guo, Y.P. Chang, L.S. Shieh, IEEE Trans. Circuits Syst. I: Regul. Pap. 54, 632 (2007)
K. Su, C. Li, Optik 126, 2671 (2015)
A. Gjurchinovski, T. Sandev, V. Urumov, J. Phys. A: Math. Theor. 43, 445102 (2010)
M.S. Tavazoei, M. Haeri, Phys. Lett. A 372, 798 (2008)
M.S. Tavazoei, M. Haeri, S. Jafari, S. Bolouki, M. Siami, IEEE Trans.Ind. Electron. 55, 4094 (2008)
M.S. Tavazoei, M. Haeri, S. Jafari, Proc. Inst. Mech. Eng. Part I: J. Syst. Control Eng. 222, 175 (2008)
E. Zambrano-Serrano, J.M. Muñoz-Pacheco, E. Campos-Cantón, Int. J. Electron. Commun. 79, 43 (2017)
M.S. Tavazoei, M. Haeri, Nonlinear Anal. Ser. B: Real World Appl. 11, 332 (2010)
W. Xingyuan, Q. Xue, Math. Prob. Eng. 2012, 601309 (2012)
I. N’Doye, K.N. Salama, T.M. Laleg-Kirati, IEEE/CAA J. Autom. Sinica 6, 268 (2019)
M. Bettayeb, U.M. Al-Saggaf, S. Djennoune, IET Control Theory Appl. 11, 3171 (2017)
M.S. Tavazoei, M. Haeri, Physica A 387, 57 (2008)
Z. Alam, L. Yuan, Q. Yang, IEEE/CAA J. Autom. Sinica 3, 157 (2016)
X.L. Gong, X.H. Liu, X. Xiong, Physica A 522, 33 (2019)
L. Chen, Y. Chai, R. Wu, Chaos 21, 043107 (2011)
A.K. Singh, V.K. Yadav, S. Das, Optik 133, 98 (2017)
M. Chen, S.Y. Shao, P. Shi, Y. Shi, IEEE Trans. Circuits Syst. II: Express Briefs 64, 417 (2017)
P. Jafari, M. Teshnehlab, M. Tavakoli-Kakhki, IET Control Theory Appl. 12, 183 (2018)
F. Meléndez-Vázquez, R. Martínez-Guerra, IET Control Theory Appl. 12, 1755 (2018)
D. Chen, R. Zhang, J.C. Sprott, H. Chen, X. Ma, Chaos 22, 023130 (2012)
X.Q. Zhang, J. Xiao, Q. Zhang, Optik 130, 1139 (2017)
Q. Zhang, J. Xiao, X.Q. Zhang, D. Cao, Optik 141, 90 (2017)
Y. Li, H. Lv, D. Jiao, AIP Adv. 7, 035106 (2017)
P. Zhoua, R. Ding, J. Franklin Inst. 348, 2839 (2011)
C.D. Cruz-Ancona, R. Martínez-Guerra, J. Franklin Inst. 354, 3054 (2017)
P. Zhou, W. Zhu, Nonlinear Anal.: Real World Appl. 12, 811 (2011)
W. Zhang, J. Cao, R. Wu, F.E. Alsaadi, A. Alsaedi, J. Franklin Inst. 356, 1522 (2019)
D.A. Yousri, A.M. Abdel, A.A. Said, A.S. Elwakil, B. Maundy, A.G. Radwan, Nonlinear Dyn. 95, 2491 (2019)
F. Liu, X. Li, Z. Liu, Y. Tang, Syst. Sci. Control Eng. 5, 42 (2016)
L. Yuan, Q. Yang, C. Zeng, Nonlinear Dyn. 73, 2491 (2013)
Y. Huang, F. Guo, Y. Li, Y. Liu, PLoS One 10, e0114910 (2015)
E. Cuevas, J. Gálvez, O. Avalos, Comput. Sistemas 21, 369 (2017)
Y. Al-Assaf, R. El-Khazali, W. Ahmad, Chaos Solitons Fractals 22, 897 (2004)
Y. Tang, X. Zhang, C. Hua, L. Li, Y. Yang, Phys. Lett. A 376, 457 (2012)
J. Lin, Z.G. Wang, Nonlinear Dyn. 90, 1243 (2017)
R.G. Li, H.N. Wu, Nonlinear Dyn. 95, 1221 (2019)
J. Liu, Z. Wang, M. Shu, F. Zhang, S. Leng, X. Sun, Complexity 2019, 7242791 (2019)
I. N’Doye, H. Voos, M. Darouach, IEEE Journal on Emerg. Sel. Top. Circuits Syst. 3, 442 (2013)
R.G. Li, H.N. Wu, Nonlinear Dyn. 92, 935 (2018)
P. Muthukumar, P. Balasubramaniam, K. Ratnavelu, Chaos 24, 033105 (2014)
P. Mani, R. Rajan, L. Shanmugam, Y.H. Joo, Inf. Sci. 491, 74 (2019)
Y.R. Bai, D. Baleanuc, G.C. Wua, Optik 168, 553 (2018)
J. Hou, R. Xi, P. Liu, T. Liu, IEEE/CAA J. Autom. Sinica 4, 381 (2017)
J.J. Montesinos-García, R. Martinez-Guerra, IET Image Process. 12, 1913 (2018)
A.J.A. El-Maksoud, A.A.A. El-Kader, B.G. Hassan, N.G. Rihan, M.F. Tolba, L.A. Said, A.G. Radwan, M.F. Abu- Elyazeed, Microelectron. J. 90, 323 (2019)
L.J. Sheu, Nonlinear Dyn. 65, 103 (2011)
Y.G. Yang, B.W. Guan, J. Li, D. Li, Y.H. Zhou, W.M. Shi, Opt. Laser Technol. 119, 105661 (2019)
M. Lahdir, H. Hamiche, S. Kassim, M. Tahanout, K. Kemih, S.A. Addouche, Opt. Laser Technol. 109, 534 (2019)
F. Tao, W. Qian, Measurement 134, 866 (2019)
J.L. Chen, C.H. Huang, Y.C. Du, C.H. Lin, IET Image Proc. 8, 354 (2014)
J.C. Sprot, Phys. Lett. A 266, 19 (2000)
K.M. Owolabi, A. Atangana, Chaos 29, 023111 (2019)
A. Atangana, J.F. Gómez-Aguilar, Chaos Solitons Fractals 14, 516 (2018)
E.F. Doungmo Goufo, M. Mbehou, M.M. Kamga Pene, Chaos Solitons Fractals 115, 170 (2018)
P.R. Nwagoum Tuwa, C.H. Miwadinou, A.V. Monwanou, J.B. Chabi Orou, P. Woafo, Mech. Res. Commun. 97, 8 (2019)
S. Faraji, M.S. Tavazoei, Cent. Eur. J. Phys. 11, 836 (2013)
H.T. Yau, S.Y. Wu, C.L. Chen, Y.C. Li, IEEE Trans. Ind. Electron. 63, 3824 (2016)
H.K. Chen, C.I. Lee, Chaos Solitons Fractals 21, 957 (2004)
H.T. Yau, C.C. Wang, C.T. Hsieh, S.Y. Wu, IET Gener. Transm. Distrib. 9, 2775 (2015)
M.H. Wang, H.T. Yau, Energies 7, 6340 (2014)
C.K. Chen, Y.C. Li, Int. J. Adv. Manuf. Technol. 100, 907 (2019)
J.C. Sprott, Phys. Lett. A 366, 397 (2007)
D.P. Wang, J.B. Yu, J. Electro. Sci. Technol. Chin. 6, 289 (2008)
M. Busłwicz, A. Makarewicz, Pomiary Automatyka Robotyka 2, 321 (2013)
R. Almeida, M. Guzowska, T. Odzijewicz, Open Math. 14, 1122 (2016)
S. Westerlund, Phys. Scr. 43, 174 (1991)
A.S. Deshpande, V. Daftardar-Gejji, Chaos Solitons Fractals 102, 119 (2017)
M.S. Tavazoei, M. Haeri, Nonlinear Dyn. 54, 213 (2008)
D. Peng, K. Sun, S. He, A.O.A. Alamodi, Chaos Solitons Fractals 119, 163 (2019)
M.S. Tavazoei, J. Comput. Nonlinear Dyn. 9, 021011 (2014)
C.P. Silva, IEEE Trans. Circuits Syst. I: Fundam. Theory App. 40, 675 (1993)
T.Y. Li, J.A. Yorke, Am. Math. Mon. 82, 985 (1975)
E. Akin, E. Glanser, W. Huang, S. Shao, X. Ye, Ergodic Theory Dyn. Syst. 30, 1277 (2010)
T. Mitra, J. Econ. Theory 96, 133 (2001)
W. Tucker, C.R. Acad. Sci., Ser. I Math. 328, 1197 (1999)
I. Stewart, Nature 406, 948 (2000)
T. Zhou, Y. Tang, G. Chen, Int. J. Bifurc. Chaos 14, 3167 (2004)
S. He, K. Sun, Y. Peng, Phys. Lett. A 383, 2267 (2019)
M.S. Tavazoei, M. Haeri, Nonlinear Dyn. 57, 363 (2009)
Z.M. Wu, J.G. Lu, J.Y. Xie, Chin. Phys. 15, 1201 (2006)
H. Zhu, Discrete Dyn. Nat. Soc. 2013, 593856 (2013)
S. Kou, K. Han, IEEE Trans. Autom. Control 20, 291 (1975)
S. Engelberg, IEEE Trans. Autom. Control 47, 1887 (2002)
V. Badri, M.S. Tavazoei, IEEE Trans. Circuits Syst. Express Briefs, https://doi.org/10.1109/TCSII.2019.2922771
P. So, Scholarpedia 2, 1353 (2007)
Y. Saiki, Nonlinear Processes Geophys. 14, 615 (2007)
E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University Press, 1993)
R. Barrio, A. Dena, W. Tucker, Comput. Phys. Commun. 194, 76 (2015)
B.M. Boghosian, A. Brown, J. Lätt, H. Tang, L.M. Fazendeiro, P.V. Coveney, Philos. Trans. R. Soc. London, Ser. A 369, 2345 (2011)
X. Yu, Y. Xia, Int. J. Bifurc. Chaos 10, 1987 (2000)
M.S. Tavazoei, M. Haeri, Automatica 45, 1886 (2009)
M.S. Tavazoei, Automatica 46, 945 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tavazoei, M.S. Fractional order chaotic systems: history, achievements, applications, and future challenges. Eur. Phys. J. Spec. Top. 229, 887–904 (2020). https://doi.org/10.1140/epjst/e2020-900238-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2020-900238-8