Abstract
In this paper, a fractional-order version of a chaotic circuit made simply of two non-idealized components operating at high frequency is presented. The fractional-order version of the Hopf bifurcation is found when the bias voltage source and the fractional-order of the system increase. Using Adams–Bashforth–Moulton predictor–corrector scheme, dynamic behaviors are displayed in two complementary types of stability diagrams, namely the two-parameter Lyapunov exponents and the isospike diagrams. The latest being a more fruitful type of stability diagrams based on counting the number of spikes contained in one period of the periodic oscillations. These two complementary types of stability diagrams are reported for the first time in the fractional-order dynamical systems. Furthermore, a new fractional-order adaptive sliding mode controller using a reduced number of control signals was built for the stabilization of a fractional-order complex dynamical network. Two examples are shown on a fractional-order complex dynamical network where the nodes are made of fractional-order two-component circuits. Firstly, we consider an ideal channel, and secondly, a non ideal one. In each case, increasing of the coupling strength leads to the phase transition in the fractional-order complex network.
Graphical abstract
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)
R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)
H. Sun, A. Abdelwahed, B. Onaral, IEEE Trans. Autom. Control 29, 441 (1984)
K. Diethelm, N.J. Ford, A.D. Freed, Nonlinear Dyn. 29, 3 (2002)
A. Atangana, I. Koca, Chaos Soliton. Fract. 89, 1 (2016)
C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu-Battle, Fractional-order Systems and Controls: Fundamentals and Applications (Springer, London, 2010)
I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulations (Springer, Berlin, Heidelberg, 2011)
J.C. Wang, J. Electrochem. Soc. 134, 1915 (1987)
R.L. Bagley, R.A. Calico, J. Guid. Control Dyn. 14, 304 (1991)
B. Ducharne, G. Sebald, D. Guyomar, G. Litak, J. Appl. Phys. 117, 243907 (2015)
W.M. Ahmad, R. El-Khazali, Chaos Soliton. Fract. 33, 1367 (2007)
L. Song, S. Xu, J. Yang, Commun. Nonlinear Sci. Numer. Simul. 15, 616 (2010)
E. Ahmed, A.S. Elgazzar, Physica A 379, 607 (2007)
B.S.T. Alkahtani, A. Atangana, Entropy 18, 100 (2016)
H.M. Baskonus, Z. Hammouch, H. Bulut, Chaos in the fractional order logistic delay system: Circuit realization and synchronization, AIP Conf. Proc. 1738, 290005 (2016)
H.M. Baskonus, T. Mekkaoui, Z. Hammouch, H. Bulut, Entropy 17, 5771 (2015)
R. Kengne, R. Tchitnga, S. Mabekou, B.R. Wafo Tekam, G.B. Soh, A. Fomethe, Chaos Soliton. Fract. 111, 6 (2018)
R. Kengne, R. Tchitnga, A. Mezatio, A. Fomethe, G. Litak, Eur. Phys. J. B 90, 88 (2017)
L.O. Chua, M. Komuro, T. Matsumoto, IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 33, 1073 (1986)
T. Hartley, C. Lorenzo, H. Qammer, IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 42, 485 (1995)
C.P. Li, W.H. Deng, D. Xu, Physica A 360, 171 (2006)
D. Cafagna, G. Grassi, Int. J. Bifurcat. Chaos 18, 615 (2008)
Z. Chaoxia, Y. Simin, Chaos Soliton. Fract. 44, 845 (2011)
C. Li, G. Chen, Chaos Soliton. Fract. 22, 540 (2004)
R. Kengne, R. Tchitnga, K.A. Tchagna, A. Fomethe, J. Eng. Sci. Technol. Rev. 6, 24 (2013)
C. Donato, G. Giuseppe, Nonlinear Dyn. 70, 1185 (2012)
R.M. Nguimdo, R. Tchitnga, P. Woafo, Chaos 23, 043122 (2013)
R. Tchitnga, H.B. Fotsin, B. Nana, P.H. Louodop, P. Woafo, Chaos Soliton. Fract. 45, 306 (2012)
J.G. Freire, J.A.C. Gallas, Chaos Soliton. Fract. 59, 129 (2014)
J.G. Freire, R. Meucci, F.T. Arecchi, J.A.C. Gallas, Chaos 25, 097607 (2015)
J.G. Freire, C. Cabeza, A.C. Marti, T.P. Oschel, J.A.C. Gallas, Eur. Phys. J. Special Topics 223, 2857 (2014)
M.R. Gallas, J.A.C. Gallas, Chaos 25, 064603 (2015)
M.R. Gallas, J.A.C. Gallas, Eur. Phys. J. Special Topics 223, 2131 (2014)
L. Junges, J.A.C. Gallas, Phys. Lett. A 376, 2109 (2012)
L. Junges, T.P. Oschel, J.A.C. Gallas, Eur. Phys. J. D 67, 149 (2013)
J. Guan, Optik 127, 4211 (2016)
A. Soukkou, A. Boukabou, S. Leulmi, Optik 127, 5070 (2016)
H. Xi, Y. Li, X. Huang, Optik 126, 5346 (2015)
R. Li, W. Li, Optik 126, 2965 (2015)
G.C. Wu, D. Baleanu, L.L. Huang, Appl. Math. Lett. 82, 71 (2018)
G.C. Wu, D. Baleanu, H.P. Xie, F.L. Chen, Physica A 460, 374 (2016)
W. Xingyuan, Z. Xiaopeng, M. Chao, Nonlinear Dyn. 69, 511 (2012)
R. Kengne, R. Tchitnga, K.A. Tchagna, T.A. Nzeusseu, A. Fomethe, J. Chaos 2013, 839038 (2013)
W. Sha, Y. Yongguang, D. Miao, Physica A 389, 4981 (2010)
D. Hongyue, Chaos Soliton. Fract. 44, 510 (2011)
X.J. Wu, H.T. Lu, Chin. Phys. B 19, 070511 (2010)
Y. Tang, J. Fang, Commun. Nonlinear Sci. Numer. Simul. 15, 401 (2010)
Y. Tang, Z. Wang, J. Fang, Chaos 19, 013112 (2009)
Z. Darui, L. Ling, L. Chongxin, Math. Probl. Eng. 2014, 936985 (2014)
W.W. Yu, G. Chen, J.H. Lu, Automatica 45, 429 (2009)
Y. Chai, L.P. Chen, R.C. Wu, J. Sun, Physica A 391, 5746 (2012)
J. Zhou, J.A. Lu, J.H. Lu, Automatica 44, 996 (2008)
H. Fotsin, S. Bowong, Chaos Soliton. Fract. 27, 822 (2006)
I. Boiko, L. Fridman, R. Iriarte, A. Pisano, E. Usai, Automatica 42, 833 (2006)
J. Slotine, W. Li, Applied Nonlinear Control (Prentice Hall, New Jersey, 1991)
J. Chen, C. Li, X. Yang, Neurocomputing 313, 324 (2018)
J. Fei, C. Lu, J. Franklin Inst. 355, 2369 (2018)
S.H. Hosseinnia, R. Ghaderi, A.N. Ranjbar, M. Mahmoudian, S. Momani, Comput. Math. Appl. 59, 1637 (2010)
H.L. Li, J. Cao, H. Jiang, A. Alsaedi, J. Franklin Inst. 335, 5771 (2018)
A. Syta, G. Litak, S. Lenci, M. Scheffler, Chaos 24, 013107 (2014)
M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, R. Castro-Linares, Commun. Nonlinear Sci. Numer. Simul. 22, 650 (2015)
A. Sharma, M.D. Shrimali, A. Prasad, R. Ramaswamy, U. Feudel, Phys. Rev. E 84, 016226 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://doi.org/creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kengne, R., Tchitnga, R., Tewa, A.K.S. et al. Fractional-order two-component oscillator: stability and network synchronization using a reduced number of control signals. Eur. Phys. J. B 91, 304 (2018). https://doi.org/10.1140/epjb/e2018-90362-7
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2018-90362-7