Mathematical views of the fractional Chua's electrical circuit described by the Caputo-Liouville derivative
DOI:
https://doi.org/10.31349/RevMexFis.67.91Keywords:
Bifurcation, Fractional Chua's electrical circuits, Lyapunov exponentAbstract
This paper revisits Chua's electrical circuit in the context of the Caputo derivative. We introduce the Caputo derivative into the modeling of the electrical circuit. The solutions of the new model are proposed using numerical discretizations. The discretizations use the numerical scheme of the Riemann-Liouville integral. We have determined the equilibrium points and study their local stability. The existence of the chaotic behaviors with the used fractional-order has been characterized by the determination of the maximal Lyapunov exponent value. The variations of the parameters of the model into the Chua's electrical circuit have been quantified using the bifurcation concept. We also propose adaptive controls under which the master and the slave fractional Chua's electrical circuits go in the same way. The graphical representations have supported all the main results of the paper.References
Cheng-Biao Fu, An-Hong Tian, Yu-Chung Li and Her-Terng Yau, Fractional Order Chaos Synchronization for Real-Time Intelligent Diagnosis of Islanding in Solar Power Grid Systems, Energies (2018), 11, 1183.
L. Chua, M. Itoh, L. Kocarev, K Eckert Chaos synchronization in Chua’s circuit , J. of Cir. Syst. and Comp. (1993), 3(1), 93-108.
M. S. Abdelouahab, and R. Lozi Bifurcation Analysis and Chaos in Simplest Fractional-order Electrical Circuit, . IEEE Proccedings of 3rd International Conference on Control, Engineering and Information Technology (CEIT) (2015), 1-5.
Ahmed J. Abd El-Maksoud et al., FPGA Implementation of Fractional-Order Chua’s Chaotic System, 7th International Conference on Modern Circuits and Systems Technologies (MOCAST) (2018).
T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, Chaos in a Fractional Order Chua’s System, IEEE Trans. on Cir. and Syst. : Fund. theo. and appli. (1995), 42(8).
N. Wang, C. Li, Generating Multi-Scroll Chua’s Attractors via Simplified Piecewise-Linear Chua’s Diode, IEEE Trans. on Cir. and Syst (2019). 21
I. Petras, Control of fractional-order Chua’s system, Journal of elect. engi. (2002), 53(7-8), 219-222.
B. S. T. Alkahtani, Chua’s circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons and Fractals (2016), 89, 547–551.
A. Atangana, and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Sci. (2016), 20(2), 763-769.
Caputo, M. and Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1(2), 1-15, (2015).
D. Avci, M. Yavuz, N. Ozdemir, Fundamental Solutions to the Cauchy and Dirichlet Problems for a Heat Conduction Equation Equipped with the Caputo-Fabrizio Differentiation, Nova Science Publishers (2019), 95-107.
A.S. Elwakil, Fractional-order circuits and systems: An emerging interdisciplinary research area, IEEE Circuits and Systems Magazine, (2010).
I. Petras, A note on the fractional-order Chua’s system, Chaos, Solitons and Fractals (2008), 38, 140-147.
Fahd, J., Abdeljawad, T., and Baleanu, D., On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10, 2607-2619, (2017).
Fahd, J. and Abdeljawad, T., A modified Laplace transform for certain generalized fractional operators, Results in Nonlinear Analysis, 2, 88-98
(2018).
K. M. Owolabi, Numerical analysis and pattern formation process for space fractional superdiffusive systems, , Disct. contin. dyn. syst. Ser. S,12(3),543-566, (2019).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands (2006), 204.
K. Saad, D. Baleanu, A. Atangana, New fractional derivatives applied to the Korteweg-de Vries and Korteweg–de Vries–Burger’s equations, Comput. Applied Math., 37(6), (2019).
N. Sene, and A. N. Fall, Homotopy Perturbation -Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation, Fractal Fract. (2019), 3, 14.
N. Sene, and J.F. Gomez-Aguilar, Analytical solutions of electrical circuits considering certain generalized fractional derivatives, Eur. Phys. J. Plus (2019) , 134: 260.
N. Sene, Integral Balance Methods for Stokes’ First, Equation Described by the Left Generalized Fractional Derivative, Physics, 1, 154-166, (2019).
N. Sene, Analytical solutions and numerical schemes of certain generalized fractional diffusion models, Eur. Phys. J. Plus, 134, 199, (2019).
N. Sene, Stability analysis of the generalized fractional differential equations with and without exogenous inputs, J. Nonlinear Sci. Appl., 12, 562-572, (2019).
N. Sene, Global asymptotic stability of the fractional differential equations, J. Nonlinear Sci. Appl., (2020), 13, 171-175.
N. Sene, Second-grade fluid model with Caputo-Liouville generalized fractional derivative, Chaos, Solitons & Fractals, 133, 109631, (2020).
N. Sene, and G. Srivastava, Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations, Symmetry (2019), 11, 608.
T. Mekkaoui, Z. Hammouch, D. Kumar and J. Singh, A New Approximation Scheme for Solving Ordinary Differential Equation with Gomez-Atangana-Caputo Fractional Derivative, Methods of Mathematical Modelling (2019), 51.
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA(1999), 198.
Z Hammouch, T Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Studies (2015), 22(4), 1-13.
J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation (2018), 316, 504-515.
Z. Hammouch, T. Mekkaoui, P. Agarwal, Optical solitons for the Calogero-Bogoyavlenskii-Schiff equation in (2+ 1) dimensions with time-fractional conformable derivative, The European Physical Journal Plus (2018), 133(7), 248.
M. F. Danca and N. Kuznetsov, Matlab Code for Lyapunov Exponents of Fractional-Order Systems, International Journal of Bifurcation and Chaos (2018), 28(5), 1850067.
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