Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method

• Autorzy: Owolabi, Kolade M.
• Języki publikacji: EN

Abstrakty

EN

In this work, synchronization of fractional dynamics of chaotic system is presented. The suggested dynamics is governed by a system of fractional differential equations, where the fractional derivative operator is modeled by the novel Caputo operator. The nature of fractional dynamical system is non-local which often rules out a closed-form solution. As a result, an efficient numerical method based on shifted Chebychev spectral collocation method is proposed. The error and convergence analysis of this scheme is also given. Numerical results are given for different values of fractional order and other parameters when applied to solve chaotic system, to address any points or queries that may occur naturally.

Słowa kluczowe

EN

Chebyshev spectral method; Caputo fractional derivative; chaotic model; numerical simulations

• Czasopismo: Journal of Applied Analysis
• Rocznik: 2021
• Tom: Vol. 27, nr 2
• Strony: 269--282
• Opis fizyczny: Bibliogr. 60 poz., wykr.

Twórcy

autor

Owolabi, Kolade M.

• Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria,
• Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa

Bibliografia

• [1] S. K. Agrawal, M. Srivastava and S. Das, Synchronization of fractional order chaotic systems using active control method, Chaos Solitons Fractals 45 (2012), 737-752.
• [2] W. M. Ahmad and W. M. Harb, On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos Solitons Fractals 18 (2003), 693-701.
• [3] W. M. Ahmad and J. C. Sprott, Chaos in fractional-order autonomous nonlinear systems, Chaos Solitons Fractals 16 (2003), 339-351.
• [4] K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, New York, 2008.
• [5] A. Atangana, A novel model for the lassa hemorrhagic fever: Deathly disease for pregnant women, Neural. Comput. Appl. 26 (2015), 1895-1903.
• [6] A. T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Cham, 2016.
• [7] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Ser. Complex. Nonlinearity Chaos 3, World Scientific, Hackensack, 2012.
• [8] B. Blasius, A. Huppert and L. Stone, Complex dynamics and phase synchronization in spatially extended ecological systems, Nature 399 (1999), 354-359.
• [9] A. Bueno-Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT 54 (2014), no. 4, 937-954.
• [10] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Astron. Soc. 13 (1967), 529-539.
• [11] A. Carpinteri, P. Cornetti and K. M. Kolwankar, Calculation of the tensile and flexural strength of disordered materials using fractional calculus, Chaos Solitons Fractals 21 (2004), 623-632.
• [12] G. Chen and T. Ueta, Yet another chaotic attractor, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 7, 1465-1466.
• [13] A. Cloot and J. F. Botha, A generalised groundwater flow equation using the concept of non-integer order derivatives, Water S. A. 32 (2006), 1-7.
• [14] K. Diethelm and N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput. 154 (2004), no. 3, 621-640.
• [15] I. Grigorenko and E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett. 91 (2003), Article ID 034101.
• [16] S. K. Han, C. Kurrer and Y. Kuramoto, Dephasing and bursting in coupled neural oscillators, Phys. Rev. Lett. 75 (1995), 3190-3193.
• [17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2001.
• [18] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006.
• [19] M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific, River Edge, 1996.
• [20] C. Li and G. Chen, Chaos and hyperchaos in the fractional-order Rössler equations, Phys. A 341 (2004), no. 1-4, 55-61.
• [21] C. Li, X. Liao and J. Yu, Synchronization of fractional order chaotic systems, Phys. Rev. E 68 (2003), Article ID 067203.
• [22] C. G. Li and G. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals 22 (2004), 549-554.
• [23] X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47 (2009), no. 3, 2108-2131.
• [24] J. G. Lu, Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos Solitons Fractals 26 (2005), 1125-1133.
• [25] R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl. 59 (2010), no. 5, 1586-1593.
• [26] R. L. Magin, Fractional calculus in bioengineering. Part 3, Crit. Rev. Biomed. Eng. 32 (2004), 195-377.
• [27] Z. Mao and J. Shen, Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients, J. Comput. Phys. 307 (2016), 243-261.
• [28] R. E. Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2005.
• [29] A. M. Mishra, S. D. Purohit, K. M. Owolabi and Y. D. Sharma, A nonlinear epidemiological model considering asymptotic and quarantine classes for SARS CoV-2 virus, Chaos Solitons Fractals 138 (2020), Article ID 109953.
• [30] R. Mittal and S. Pandit, Quasilinearized Scale-3Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems, Eng. Computation 35 (2018), 1907-1931.
• [31] J. D. Murray, Mathematical Biology. I: An Introduction, Springer, New York, 2003.
• [32] J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, Springer, New York, 2003.
• [33] P. A. Naik, J. Zu and K. M. Owolabi, Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order, Phys. A 545 (2020), Article ID 123816.
• [34] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover, New York, 2006.
• [35] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer, New York, 2011.
• [36] K. M. Owolabi, Efficient Numerical Methods for Reaction-Diffusion Problems, VDM, Saarbrücken, 2016.
• [37] K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul. 44 (2017), 304-317.
• [38] K. M. Owolabi, Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann-Liouville derivative, Numer. Methods Partial Differential Equations 34 (2018), no. 1, 274-295.
• [39] K. M. Owolabi, Riemann-Liouville fractional derivative and application to model chaotic differential equations, Progr. Fract. Differ. Appl. 4 (2018), 99-110.
• [40] K. M. Owolabi, Mathematical modelling and analysis of love dynamics: A fractional approach, Phys. A 525 (2019), 849-865.
• [41] K. M. Owolabi, Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann-Liouville derivative, Progr. Fract. Differ. Appl. 6 (2020), 1-14.
• [42] K. M. Owolabi and A. Atangana, Chaotic behaviour in system of noninteger-order ordinary differential equations, Chaos Solitons Fractals 115 (2018), 362-370.
• [43] K. M. Owolabi and A. Atangana, Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction—diffusion systems, Comput. Appl. Math. 37 (2018), no. 2, 2166-2189.
• [44] K. M. Owolabi and A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Chaos 29 (2019), no. 2, Article ID 023111.
• [45] K. M. Owolabi, J. F. Gómez-Aguilar and B. Karaagac, Modelling, analysis and simulations of some chaotic systems using derivative with Mittag-Leffler kernel, Chaos Solitons Fractals 125 (2019), 54-63.
• [46] J. H. Park and O. M. Kwon, A novel criterion for delayed feedback control of time-delay chaotic systems, Chaos Solitons Fractals 23 (2005), no. 2, 495-501.
• [47] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990), no. 8, 821-824.
• [48] I. Petrás, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, Berlin, 2011.
• [49] E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simul. 40 (2016), 112-128.
• [50] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [51] A. G. Radwan, K. Moaddy and S. Momani, Stability and non-standard finite difference method of the generalized Chua’s circuit, Comput. Math. Appl. 62 (2011), no. 3, 961-970.
• [52] A. G. Radwan, K. Moaddy, K. N. Salama, S. Momani and I. Hashim, Control and switching synchronization of fractional order chaotic systems using active control technique, J. Adv. Res. 5 (2014), 125-132.
• [53] M. A. Snyder, Chebyshev Methods in Numerical Approximation, Prentice-Hall, Englewood Cliffs, 1966.
• [54] E. Sousa, Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys. 228 (2009), no. 11, 4038-4054.
• [55] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer, New York, 1982.
• [56] J. C. Strikwerda, Partial Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2004.
• [57] D. Tripathi, S. K. Pandey and S. Das, Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel, Appl. Math. Comput. 215 (2010), no. 10, 3645-3654.
• [58] T. Ueta and G. Chen, Bifurcation analysis of Chen’s equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 (2000), no. 8, 1917-1931.
• [59] Q. Xu and J. S. Hesthaven, Stable multi-domain spectral penalty methods for fractional partial differential equations, J. Comput. Phys. 257 (2014), 241-258.
• [60] Y. Yu and H.-X. Li, The synchronization of fractional-order Rössler hyperchaotic systems, Phys. A 387 (2008), no. 5-6, 1393-1403.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021)

Identyfikatory

DOI

10.1515/jaa-2021-2053