| Citation: |
S. M. Sohel Rana, Md. Jasim Uddin, A. Q. Khan. BIFURCATION AND CHAOS IN A DISCRETE FRACTIONAL ORDER REDUCED LORENZ MODEL WITH CAPUTO AND CONFORMABLE DERIVATIVES[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1241-1271. doi: 10.11948/20240181
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BIFURCATION AND CHAOS IN A DISCRETE FRACTIONAL ORDER REDUCED LORENZ MODEL WITH CAPUTO AND CONFORMABLE DERIVATIVES
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Abstract
This study investigates a discrete fractional order reduced Lorenz model that incorporates the Caputo and Conformable fractional derivatives respectively. The stability of equilibrium points of the model with Caputo fractional derivative are analyzed. The Conformable fractional derivative model is likewise examined. We confirm algebraically the existence and direction of Neimark-Sacker bifurcation for both models using the central manifold and bifurcation theories. The dynamic behavior of these models have been extensively investigated based on changes made to the control parameters. In addition to supporting analytical findings, numerical simulations are used to reveal chaotic characteristics such as bifurcations, phase portraits, periodic orbits, invariant closed cycles, and attractive chaotic sets. We also quantitatively compute the maximal Lyapunov exponents and fractal dimensions to validate the chaotic properties of the system. Using three different control strategies viz, OGY, hybrid control method, and state feedback method, the system's chaotic trajectory is finally stopped.
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References
[1] M. A. M. Abdelaziz, A. I. Ismail, F. A. Abdullah and M. H. Mohd, Codimension one and two bifurcations of a discrete-time fractional-order SEIR measles epidemic model with constant vaccination, Chaos, Solitons & Fractals, 2020, 140, 110104. [2] M. A. M. Abdelaziz, A. I. Ismail, F. Abdullah and M. H. Mohd, Bifurcations and chaos in a discrete SI epidemic model with fractional order, Advances in Difference Equations, 2018, 2018(1), 1–19. doi: 10.1186/s13662-017-1452-3 [3] R. P. Agarwal, A. El-Sayed and S. M. Salman, Fractional-order Chua's system: Discretization, bifurcation and chaos, Advances in Difference Equations, 2013, 1(2013), 1–13. [4] H. N. Agiza, E. M. Elabbasy, H. El-Metwally and A. A. Elsadany, Chaotic dynamics of a discrete prey–predator model with Holling type Ⅱ, Nonlinear Analysis: Real World Applications, 2009, 10(1), 116–129. doi: 10.1016/j.nonrwa.2007.08.029 [5] W. M. Ahmad and J. C. Sprott, Chaos in fractional-order autonomous nonlinear systems, Chaos, Solitons & Fractals, 2003, 16(2), 339–351. [6] R. Ahmed, N. Tahir and N. A. Shah, An analysis of the stability and bifurcation of a discrete-time predator–prey model with the slow–fast effect on the predator, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2024, 34, 3. [7] S. Akhtar, R. Ahmed, M. Batool, N. A. Shah and J. D. Chung, Stability, bifurcation and chaos control of a discretized Leslie prey-predator model, Chaos, Solitons & Fractals, 2021, 152, 111345. [8] I. Ameen and P. Novati, The solution of fractional order epidemic model by implicit Adams methods, Applied Mathematical Modelling, 2017, 43, 78–84. doi: 10.1016/j.apm.2016.10.054 [9] A. Atangana, A novel model for the lassa hemorrhagic fever: Deathly disease for pregnant women, Neural Computing and Applications, 2015, 26(8), 1895–1903. doi: 10.1007/s00521-015-1860-9 [10] R. L. Bagley and R. Calico, Fractional order state equations for the control of visco–elastically damped structures, Journal of Guidance, Control, and Dynamics, 1991, 14(2), 304–311. doi: 10.2514/3.20641 [11] E. Balcı, S. Kartal and İ Öztürk, Comparison of dynamical behavior between fractional order delayed and discrete conformable fractional order tumor-immune system, Mathematical Modelling of Natural Phenomena, 2021, 16(3). [12] E. Balcı, İ Öztürk and S. Kartal, Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative, Chaos, Solitons & Fractals, 2019, 123, 43–51. [13] D. Baleanu, A. Jajarmi, E. Bonyah and M. Hajipour, New aspects of poor nutrition in the life cycle within the fractional calculus, Advances in Difference Equations, 2018, 2018(1), 1–14. doi: 10.1186/s13662-017-1452-3 [14] N. S. S. Barhoom and S. Al-Nassir, Discretization fractional-order biological model with optimal harvesting, Abstract and Applied Analysis, 2022, 2022. DOI: 10.1155/2022/8171037 .[15] L. Bolton, A. Cloot, H. Alain, S. W. Schoombie, J. Slabbert and P. Jacobus, A proposed fractional-order Gompertz model and its application to tumour growth data, Mathematical Medicine and Biology: A Journal of the IMA, 2015, 32(2), 187–209. doi: 10.1093/imammb/dqt024 [16] J. H. E. Cartwright, Nonlinear stiffness, Lyapunov exponents, and attractor dimension, Physics Letters A, 1999, 264(4), 298–302. doi: 10.1016/S0375-9601(99)00793-8 [17] W. S. Chung, Fractional Newton mechanics with conformable fractional derivative, Journal of Computational and Applied Mathematics, 2015, 290, 150–158. doi: 10.1016/j.cam.2015.04.049 [18] Y. Cui, H. He, G. Sun and C. Lu, Analysis and control of fractional order generalized Lorenz chaotic system by using finite time synchronization, Adv. Math. Phys., 2019, 1–12. DOI: 10.1155/2019/3713789 .[19] O. Edward, G. Celso and A. Y. James, Controlling chaos, Physical Review Letters, 1990, 64(11), 1196–1199. doi: 10.1103/PhysRevLett.64.1196 [20] E. M. Elabbasy, A. A. Elsadany and Y. Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Applied Mathematics and Computation, 2014, 228, 184–194. doi: 10.1016/j.amc.2013.11.088 [21] A. A. Elsadany and A. E. Matouk, Dynamical behaviors of fractional-order Lotka–Volterra predator–prey model and its discretization, Journal of Applied Mathematics and Computing, 2015, 49(1), 269–283. [22] A. Elsonbaty and A. A. Elsadany, Bifurcation analysis of chaotic geomagnetic field model, Chaos, Solitons & Fractals, 2017, 103, 325–335. [23] Z. Eskandari, P. A. Naik and M. Yavuz, Dynamical behaviors of a discrete-time prey–predator model with harvesting effect on the predator, J. Appl. Anal. Comput, 2024, 14, 283–297. [24] C. E. Frouzakis, I. G. Kevrekidis and B. B. Peckham, A route to computational chaos revisited: Noninvertibility and the breakup of an invariant circle, Physica D: Nonlinear Phenomena, 2003, 177(1–4), 101–121. doi: 10.1016/S0167-2789(02)00751-0 [25] Y. Gao annd B. Liu, Study on the dynamical behaviors of a two-dimensional discrete system, Nonlinear Analysis: Theory, Methods & Applications, 2009, 70(12), 4209–4216. [26] Y. Gu, G. Li, X. Xu, X. Song and H. Zhong, Solution of a new high-performance fractional-order Lorenz system and its dynamics analysis, Nonlinear Dynamics, 2023, 111(8), 7469–7493. doi: 10.1007/s11071-023-08239-7 [27] Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response, Nonlinear Analysis: Real World Applications, 2011, 12(4), 2356–2377. [28] C. Huang, J. Cao, M. Xiao, A. Alsaedi and F. E. Alsaadi, Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders, Applied Mathematics and Computation, 2017, 293, 293–310. [29] C. Huang, J. Cao, M. Xiao, A. Alsaedi and T. Hayat, Bifurcations in a delayed fractional complex-valued neural network, Applied Mathematics and Computation, 2017, 292, 210–227. [30] C. Huang, J. Cao, M. Xiao, A. Alsaedi and T. Hayat, 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, 2018, 57, 1–13. [31] C. Huang, Y. Meng, J. Cao, A. Alsaedi and F. E. Alsaadi, New bifurcation results for fractional BAM neural network with leakage delay, Chaos, Solitons & Fractals, 2017, 100, 31–44. [32] M. Ichise, Y. Nagayanagi and T. Kojima, An analog simulation of non-integer order transfer functions for analysis of electrode processes, Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 1971, 33(2), 253–265. [33] H. Jafari and V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, Journal of Computational and Applied Mathematics, 2006, 196(2), 644–651. [34] A. Jajarmi and D. Baleanu, A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems, Frontiers in Physics, 2020, 8, 220. [35] Z. Jing and J. Yang, Bifurcation and chaos in discrete-time predator–prey system, Chaos, Solitons & Fractals, 2006, 27(1), 259–277. [36] S. Kartal and F. Gurcan, Discretization of conformable fractional differential equations by a piecewise constant approximation, International Journal of Computer Mathematics, 2019, 96(6), 1849–1860. [37] R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 2014, 264, 65–70. DOI: 10.1016/j.cam.2014.01. [38] A. Q. Khan, S. A. H. Bukhari and M. B. Almatrafi, Global dynamics, Neimark-Sacker bifurcation and hybrid control in a Leslie's prey-predator model, Alexandria Engineering Journal, 2022, 61(12), 11391–11404. [39] A. Q. Khan and T. Khalique, Neimark-Sacker bifurcation and hybrid control in a discrete-time Lotka-Volterra model, Mathematical Methods in the Applied Sciences, 2020, 43(9), 5887–5904. [40] E. N. Lorenz, Computational chaos-a prelude to computational instability, Physica D: Nonlinear Phenomena, 1989, 35(3), 299–317. [41] C. Luo and X. Wang, Chaos in the fractional-order complex Lorenz system and its synchronization, Nonlinear Dynamics, 2013, 71, 241–257. [42] S. Lynch and others, Dynamical Systems with Applications using Mathematica, Springer, 2007. [43] S. Momani and Z. Odibat, Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics, 2007, 207(1), 96–110. [44] D. Mozyrska and W. Małgorzata, Stability analysis of discrete-time fractional-order Caputo-type Lorenz system, 42nd International Conference on Telecommunications and Signal Processing (TSP), IEEE, 2019, 149–153. DOI: 10.1109/TSP.2019.8768861. [45] P. A. Naik, Z. Eskandari, H. E. Shahkari and K. M. Owolabi, Bifurcation analysis of a discrete-time prey-predator model, Bulletin of Biomathematics, 2023, 1(2), 111–123. [46] P. A. Naik, Z. Eskandari, M. Yavuz and J. Zu, Complex dynamics of a discrete-time Bazykin–Berezovskaya prey-predator model with a strong Allee effect, Journal of Computational and Applied Mathematics, 2022, 413, 114401. [47] C. M. A. Pinto and J. T. Machado, Fractional model for malaria transmission under control strategies, Computers & Mathematics with Applications, 2013, 66(5), 908–916. [48] I. Podlubny, Fractional Differential Equations, NewYork: Academic Press, 1999. [49] Z. F. El Raheem and S. M. Salman, On a discretization process of fractional-order logistic differential equation, Journal of the Egyptian Mathematical Society, 2014, 22(3), 407–412. [50] S. M. Rana, Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model, Journal of the Egyptian Mathematical Society, 2019, 27(1), 1–16. [51] S. M. Rana and U. Kulsum, Bifurcation analysis and chaos control in a discrete-time predator-prey system of Leslie type with simplified Holling type Ⅳ functional response, Discrete Dynamics in Nature and Society, 2017, 2017. DOI: 10.1155/2017/9705985. [52] S. M. S. Rana and M. J. Uddin, Dynamics of a discrete-time chaotic Lü system, Pan-American Journal of Mathematics, 2022, 1, 7. DOI: 10.28919/cpr-pajm/1-7. [53] M. Richard, D. Manuel, P. Igor and T. Juan, On the fractional signals and systems, Signal Processing, 2011, 91(3), 350–371. [54] S. M. Salman, A. M. Yousef and A. A. Elsadany, Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response, Chaos, Solitons & Fractals, 2016, 93, 20–31. [55] C. Sparrow, The Lorenz Equations, Springer, New York, 1982, 269. [56] N. Tahir, R. Ahmed and N. A. Shah, Complex dynamics of a discrete-time Leslie–Gower predator–prey system with herd behavior and slow–fast effect on predator population, International Journal of Biomathematics, 2024, 2450040, 26. [57] M. J. Uddin, M. B. Almatrafi and S. Monira, A study on qualitative analysis of a discrete fractional order prey-predator model with intraspecific competition, Commun. Math. Biol. Neurosci., 2024, 2024. DOI: 10.28919/cmbn/8477. [58] M. J. Uddin and S. M. S. Rana, Analysis of chaotic dynamics: A fractional order glycolysis model, Network Biology, 2022. http://www.iaees.org/publications/journals/nb/articles/2022-12(4)/2022-12(4).asp .[59] M. J. Uddin and S. M. Sohel Rana, Qualitative analysis of the discretization of a continuous fractional order prey-predator model with the effects of harvesting and immigration in the population, Complexity, 2024, 2024(1). DOI: 10.1155/2024/8855142. [60] M. J. Uddin, S. M. S. Rana, S. Işık and F. Kangalgil, On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population, Chaos, Solitons & Fractals, 2023, 175, 113932. [61] X. Xie, L. Zujun and W. Hui, An image information hiding algorithm based on chaotic permutation, NJ Inf. Secur. Commun., 2007, 6, 187–191. [62] Y. Xie, A fractional‐order discrete Lorenz map, Advances in Mathematical Physics, 2022, 1, 2881207. [63] Y. Xu, R. Gu, H. Zhang and D. Li, Chaos in diffusionless Lorenz system with a fractional order and its control, International Journal of Bifurcation and Chaos, 2012, 22(04), 1250088. [64] L. G. Yuan and Q. G. Yang, Bifurcation, invariant curve and hybrid control in a discrete-time predator–prey system, Applied Mathematical Modelling, 2015, 39(8), 2345–2362. -
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S. M. Sohel Rana, Md. Jasim Uddin, A. Q. Khan. BIFURCATION AND CHAOS IN A DISCRETE FRACTIONAL ORDER REDUCED LORENZ MODEL WITH CAPUTO AND CONFORMABLE DERIVATIVES[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1241-1271. doi: 10.11948/20240181
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Figure 1.
Bifurcation diagram of system (2.3) and (3.1) at the equilibrium point
$ E_{+} $ $ b=1, \alpha_{d}=0.58 $ -
Figure 2.
(ⅰ) Neimark-Sacker bifurcation in
$ x $ $ y $ $ a=0.5, b=1.0, \alpha_{d}= 0.58, $ $ \rho_{d} $ $ 0.7 \leq \rho_{d} \leq 2.2 $ $ (x^{*}, y^{*})=(0.707107, 0.5). $ -
Figure 3.
Phase portraits for various values of
$ \rho_{d} \quad ( \rho_{d}=0.75, 0.85, 1.24, 1.453, 1.465, 1.6, 1.69, 2.2 ) $ -
Figure 4.
Neimark-Sacker bifurcation of system (2.3) in (ⅰ-ⅲ)
$ x-\rho_{d} $ $ \alpha_{d}<1 $ $ \alpha_{d}=1 $ $ x-a $ -
Figure 5.
(ⅰ) Neimark-Sacker bifurcation in
$ x $ $ y $ $ a=0.5, b=1.0, \rho_{d}= 0.820228, $ $ \alpha_{d} $ $ 0.01 \leq \alpha_{d} \leq 0.58 $ $ (x^{*}, y^{*})=(0.707107, 0.5) $ -
Figure 6.
(ⅰ) Neimark-Sacker bifurcation in
$ x $ $ y $ $ a=0.5, b=1.0, \alpha_{d}= 0.58, $ $ \rho_{d} $ $ 0.3 \leq \rho_{d} \leq 4.5 $ $ (x^{*}, y^{*})=(0.707107, 0.5) $ -
Figure 7.
Phase portraits for various values of
$ \rho_{d} \quad ( \rho_{d}= 0.37, 0.39095, 0.42, 1.14, 2, 4.5 ) $ -
Figure 8.
$ 0-1 $ $ \rho $ $ s $ $ (\rho,K) $ -
Figure 9.
Stable region and trajectory of controlled Caputo reduced Lorenz system, (ⅰ-ⅱ) OGY approach system
$ (6.3) $ $ (6.8) $ $ (6.9) $ -
Figure 10.
Neimark-Sacker bifurcation of the control system
$ (6.8) $ $ x $ $ \omega_e=0.85 $ $ \omega_e=0.75 $ $ \omega_e=0.65 $ $ \omega_e=0.58. $ -
Figure 11.
Stable region and trajectory of controlled conformable reduced Lorenz system.