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

References

[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. 10.1016/j.chaos.2012.02.004Search in Google Scholar

[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. 10.1016/S0960-0779(02)00644-6Search in Google Scholar

[3] W. M. Ahmad and J. C. Sprott, Chaos in fractional-order autonomous nonlinear systems, Chaos Solitons Fractals 16 (2003), 339–351. 10.1016/S0960-0779(02)00438-1Search in Google Scholar

[4] K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, New York, 2008. Search in Google Scholar

[5] A. Atangana, A novel model for the lassa hemorrhagic fever: Deathly disease for pregnant women, Neural. Comput. Appl. 26 (2015), 1895–1903. 10.1007/s00521-015-1860-9Search in Google Scholar

[6] A. T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Cham, 2016. 10.1007/978-3-319-30340-6Search in Google Scholar

[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. 10.1142/8180Search in Google Scholar

[8] B. Blasius, A. Huppert and L. Stone, Complex dynamics and phase synchronization in spatially extended ecological systems, Nature 399 (1999), 354–359. 10.1038/20676Search in Google Scholar PubMed

[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. 10.1007/s10543-014-0484-2Search in Google Scholar

[10] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Astron. Soc. 13 (1967), 529–539. 10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[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. 10.1016/j.chaos.2003.12.081Search in Google Scholar

[12] G. Chen and T. Ueta, Yet another chaotic attractor, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 7, 1465–1466. 10.1142/S0218127499001024Search in Google Scholar

[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. 10.4314/wsa.v32i1.5225Search in Google Scholar

[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. 10.1016/S0096-3003(03)00739-2Search in Google Scholar

[15] I. Grigorenko and E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett. 91 (2003), Article ID 034101. 10.1103/PhysRevLett.91.034101Search in Google Scholar PubMed

[16] S. K. Han, C. Kurrer and Y. Kuramoto, Dephasing and bursting in coupled neural oscillators, Phys. Rev. Lett. 75 (1995), 3190–3193. 10.1103/PhysRevLett.75.3190Search in Google Scholar PubMed

[17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2001. 10.1142/3779Search in Google Scholar

[18] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006. Search in Google Scholar

[19] M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific, River Edge, 1996. 10.1142/2637Search in Google Scholar

[20] C. Li and G. Chen, Chaos and hyperchaos in the fractional-order Rössler equations, Phys. A 341 (2004), no. 1–4, 55–61. 10.1016/j.physa.2004.04.113Search in Google Scholar

[21] C. Li, X. Liao and J. Yu, Synchronization of fractional order chaotic systems, Phys. Rev. E 68 (2003), Article ID 067203. 10.1103/PhysRevE.68.067203Search in Google Scholar PubMed

[22] C. G. Li and G. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals 22 (2004), 549–554. 10.1016/j.chaos.2004.02.035Search in Google Scholar

[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. 10.1137/080718942Search in Google Scholar

[24] J. G. Lu, Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos Solitons Fractals 26 (2005), 1125–1133. 10.1016/j.chaos.2005.02.023Search in Google Scholar

[25] R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl. 59 (2010), no. 5, 1586–1593. 10.1016/j.camwa.2009.08.039Search in Google Scholar

[26] R. L. Magin, Fractional calculus in bioengineering. Part 3, Crit. Rev. Biomed. Eng. 32 (2004), 195–377. 10.1615/CritRevBiomedEng.v32.i34.10Search in Google Scholar PubMed

[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. 10.1016/j.jcp.2015.11.047Search in Google Scholar

[28] R. E. Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2005. 10.1142/5884Search in Google Scholar

[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. 10.1016/j.chaos.2020.109953Search in Google Scholar PubMed PubMed Central

[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. 10.1108/EC-09-2017-0347Search in Google Scholar

[31] J. D. Murray, Mathematical Biology. I: An Introduction, Springer, New York, 2003. 10.1007/b98869Search in Google Scholar

[32] J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, Springer, New York, 2003. 10.1007/b98869Search in Google Scholar

[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. 10.1016/j.physa.2019.123816Search in Google Scholar

[34] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover, New York, 2006. Search in Google Scholar

[35] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer, New York, 2011. 10.1007/978-94-007-0747-4Search in Google Scholar

[36] K. M. Owolabi, Efficient Numerical Methods for Reaction-Diffusion Problems, VDM, Saarbrücken, 2016. Search in Google Scholar

[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. 10.1016/j.cnsns.2016.08.021Search in Google Scholar

[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. 10.1002/num.22197Search in Google Scholar

[39] K. M. Owolabi, Riemann–Liouville fractional derivative and application to model chaotic differential equations, Progr. Fract. Differ. Appl. 4 (2018), 99–110. 10.18576/pfda/040204Search in Google Scholar

[40] K. M. Owolabi, Mathematical modelling and analysis of love dynamics: A fractional approach, Phys. A 525 (2019), 849–865. 10.1016/j.physa.2019.04.024Search in Google Scholar

[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. 10.1002/num.22197Search in Google Scholar

[42] K. M. Owolabi and A. Atangana, Chaotic behaviour in system of noninteger-order ordinary differential equations, Chaos Solitons Fractals 115 (2018), 362–370. 10.1016/j.chaos.2018.07.034Search in Google Scholar

[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. 10.1007/s40314-017-0445-xSearch in Google Scholar

[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. 10.1063/1.5085490Search in Google Scholar PubMed

[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. 10.1016/j.chaos.2019.05.019Search in Google Scholar

[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. 10.1016/j.chaos.2004.05.023Search in Google Scholar

[47] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990), no. 8, 821–824. 10.1016/B978-012396840-1/50040-0Search in Google Scholar

[48] I. Petrás, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, Berlin, 2011. 10.1007/978-3-642-18101-6Search in Google Scholar

[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. 10.1016/j.cnsns.2016.04.020Search in Google Scholar

[50] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Search in Google Scholar

[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. 10.1016/j.camwa.2011.04.047Search in Google Scholar

[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. 10.1016/j.jare.2013.01.003Search in Google Scholar PubMed PubMed Central

[53] M. A. Snyder, Chebyshev Methods in Numerical Approximation, Prentice-Hall, Englewood Cliffs, 1966. Search in Google Scholar

[54] E. Sousa, Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys. 228 (2009), no. 11, 4038–4054. 10.1016/j.jcp.2009.02.011Search in Google Scholar

[55] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer, New York, 1982. 10.1007/978-1-4612-5767-7Search in Google Scholar

[56] J. C. Strikwerda, Partial Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2004. 10.1137/1.9780898717938Search in Google Scholar

[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. 10.1016/j.amc.2009.11.002Search in Google Scholar

[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. 10.1142/S0218127400001183Search in Google Scholar

[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. 10.1016/j.jcp.2013.09.041Search in Google Scholar

[60] Y. Yu and H.-X. Li, The synchronization of fractional-order Rössler hyperchaotic systems, Phys. A 387 (2008), no. 5–6, 1393–1403. 10.1016/j.physa.2007.10.052Search in Google Scholar