Journal of Applied Mathematics and Computation

 ISSN Print: 2576-0645 Downloads: 99868 Total View: 1476061 Frequency: quarterly ISSN Online: 2576-0653 CODEN: JAMCEZ Email: jamc@hillpublisher.com

Volumes & Issues

Current Issue

Article http://dx.doi.org/10.26855/jamc.2022.09.008

Analysis of Bifurcation and chaos to Fractional order Brusselator Model

Md. Jasim Uddin

Department of Mathematics, University of Dhaka, Dhaka, Bangladesh.

*Corresponding author: Md. Jasim Uddin

Published: September 21,2022

Abstract

The Caputo fractional derivative has been considered the Brusselator model. A discretization procedure is initially used to construct caputo fractional differential equations for Brusselator model. We list the topological categories for this model fixed points. Then, we demonstrate analytically that a fractional order Brusselator model underpins a Neimark-Sacker (NS) bifurcation and a Flip-bifurcation under specific parametric conditions. We establish the existence and direction of NS and Flip bifurcations by employing central manifold and bifurcation theory. The dynamical behavior of the fractional order Brusselator model has been determined to be extremely sensitive to the parameter values and the initial conditions. It is investigated how the model's dynamics are affected by step size and fractional- order parameters. We run numerical simulations to support our analytic results, including bifurcations, phase portraits, periodic orbits, invariant closed cycles, rapid emergence of chaos, and abrupt removal of chaos. Finally, a hybrid control method is used to stop the systems chaotic trajectory.

References

[1] Kuznetsov, Y.A., Piccardi, C. (1994). Bifurcation analysis of periodic SEIR and SIR epidemic models. J. Math. Biol. 32, 109-121.

[2] Hethcote, H.W., Driessche, P.V. (1995). An SIS epidemic model with variable population size and delay. J. Math. Biol. 34, 177-194.

[3] Richard Magin, Manuel D. Ortigueira, Igor Podlubny, Juan Trujillo. (2011). On the fractional signals and systems. 91(3), 350-371.

[4] Chengdai Huang, Jinde Cao, Min Xiao, Ahmed Alsaedi, Fuad E. Alsaadi. (2017). Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders. Applied Mathematics and Computation, 293, 293-310.

[5] Chengdai Huang, Jinde Cao, Min Xiao, Ahmed Alsaedi, Tasawar Hayat. (2017). Bifurcations in a delayed fractional complex-valued neural network. Applied Mathematics and Computation, 292, 210-227.

[6] Chengdai Huang, Jinde Cao, Min Xiao, Ahmed Alsaedi, Tasawar Hayat. (2018). Effects of time delays on stability and Hopf bifurcation in a fractional ring-structured network with arbitrary neurons. Communications in Nonlinear Science and Numerical Simulation, 57, 1-13.

[7] Chengdai Huang, Yijie Meng, Jinde Cao, Ahmed Alsaedi, Fuad E. Alsaadi. (2017). New bifurcation results for fractional BAM neural network with leakage delay. Chaos, Solitons & Fractals, 100, 31-44.

[8] Al-Khaled, K., Alquran, M. (2014). An approximate solution for a fractional model of generalized Harry Dym equation. J. Math. Sci. 8, 125-130.

[9] Bagley, R. L. and Calico, R. (1991). Fractional order state equations for the control of visco elastically damped structures. Journal of Guidance, Control, and Dynamics, 14(2), 304-311.

[10] Ichise, M., Nagayanagi, Y., Kojima, T. (1971). An analog simulation of non-integer order transfer functions for analysis of electrode process. J. Electroanal. Chem. Interfacial Electrochem. 33, 253-265.

[11] Ahmad, W.M., Sprott, J.C. (2003). Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16, 339-351.

[12] Podlubny, I. (1999). Fractional Differential Equations. New York: Academic Press.

[13] Hossein Jafari, Varsha Daftardar-Gejji. (2006). Solving a system of nonlinear fractional differential equations using Adomian decomposition. Journal of Computational and Applied Mathematics, 196(2), 644-651.

[14] I. Ameen, P. Novati. (2017). The solution of fractional order epidemic model by implicit Adams methods. Applied Mathematical Modelling, 43, 78-84.

[15] Shaher Momani, Zaid Odibat. (2007). Numerical approach to differential equations of fractional order. Journal of Computational and Applied Mathematics, 96-110.

[16] Elsadany, A. A., Matouk, A. E. (2015). Dynamical behaviors of fractional-order Lotka–Volterra predator–prey model and its discretization. Journal of Applied Mathematics and Computing, 49, 269-283.

[17] Ercan Balci, Senol Kartal, Ilhan Ozturk. (2021). Comparison of dynamical behavior between fractional order delayed and discrete conformable fractional order tumor-immune system. Math. Model. Nat. Phenom, 16(3).

[18] Ercan Balcı, İlhan Öztürk, Senol Kartal. (2019). Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative. Chaos, Solitons & Fractals, 123, 43-51.

[19] Abdelaziz, M.A.M., Ismail, A.I., Abdullah, F.A. et al. (2018). Bifurcations and chaos in a discrete SI epidemic model with fractional order. Adv Differ Equ, 2018(44).

[20] Khan, Abdul Qadeer & Khalique, Tanzeela. (2020). Neimark-Sacker bifurcation and hybrid control in a discrete-time Lotka‐Volterra model. Mathematical Methods in the Applied Sciences, 43(9), 5887-5904.

[21] Khan, AQ, Bukhari, SAH & Almatrafi, MB. (2022). Global dynamics, Neimark-Sacker bifurcation and hybrid control in a Leslie’s prey-predator model Global dynamics, Neimark-Sacker bifurcation and hybrid control in a Leslie’s prey-predator model. Alexandria Engineering Journal, 61(12), 11391-11404.

[22] I. Prigogine, R. Lefever. (1968) Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695-1700.

[23] G. Nicolis, I. Prigogine. (1977). Self-Organizations in Non-equilibrium Systems (Wiley-Interscience, New York).

[24] Kuznetsov, Y. (1998). Elements of applied bifurcation theory (Vol. 112). New York, USA: Springer Science and Business Media.

[25] Wen, G. (2005). Criterion to identify hopf bifurcations in maps of arbitrary dimension. Physical Review E, 72(2), 026201.

[26] Yao, S. (2012). New Bifurcation Critical Criterion of Flip-Neimark-Sacker Bifurcations for Two-Parameterized Family of n -Dimensional Discrete Systems. Discrete Dynamics in Nature and Society, 2012, 264526.

[27] Yuan, L. G., Yang, Q. G. (2015). Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system. Applied Mathematical Modelling, 39(8), 2345-2362.

[28] S.H. Strogatz. (1994). Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, New York).

[29] X.S. Luo, G.R. Chen. (2003). B.H.Wang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems. Chaos Soliton Fractals 18, 775-783.

[30] J.L. Ren, L.P. Yu. (2016). Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model. J. NonlinearSci. 26, 1895-1931.

How to cite this paper

Analysis of Bifurcation and chaos to Fractional order Brusselator Model

How to cite this paper:  Md. Jasim Uddin. (2022) Analysis of Bifurcation and chaos to Fractional order Brusselator Model. Journal of Applied Mathematics and Computation6(3), 356-369.

DOI: http://dx.doi.org/10.26855/jamc.2022.09.008