Dynamical analysis and boundedness for a generalized chaotic Lorenz model

Xinna Mao; Hongwei Feng; Maryam A. Al-Towailb; Hassan Saberi-Nik; Xinna Mao; Hongwei Feng; Maryam A. Al-Towailb; Hassan Saberi-Nik

doi:10.3934/math.20231005

AIMS Mathematics

2023, Volume 8, Issue 8: 19719-19742. doi: 10.3934/math.20231005

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Research article

Dynamical analysis and boundedness for a generalized chaotic Lorenz model

1.
School of Mathematics and Statistics, Xinxiang University, Xinxiang, Henan 453000, China
2.
School of Civil Engineering and Architecture, Xinxiang University, Xinxiang, Henan 453000, China
3.
Department of Computer Science and Engineering, College of Applied Studies and Community Service, King Saud University, Riyadh, KSA
4.
Department of Mathematics and Statistics, University of Neyshabur, Neyshabur, Iran

Received: 06 April 2023 Revised: 23 May 2023 Accepted: 05 June 2023 Published: 12 June 2023
MSC : 34H10, 34H15, 34H20

The dynamical behavior of a 5-dimensional Lorenz model (5DLM) is investigated. Bifurcation diagrams address the chaotic and periodic behaviors associated with the bifurcation parameter. The Hamilton energy and its dependence on the stability of the dynamical system are presented. The global exponential attractive set (GEAS) is estimated in different 3-dimensional projection planes. A more conservative bound for the system is determined, that can be applied in synchronization and chaos control of dynamical systems. Finally, the finite time synchronization of the 5DLM, indicating the role of the ultimate bound for each variable, is studied. Simulations illustrate the effectiveness of the achieved theoretical results.
- generalized chaotic Lorenz system,
- globally exponentially attractive set,
- ultimate bound set,
- finite-time synchronization
Citation: Xinna Mao, Hongwei Feng, Maryam A. Al-Towailb, Hassan Saberi-Nik. Dynamical analysis and boundedness for a generalized chaotic Lorenz model[J]. AIMS Mathematics, 2023, 8(8): 19719-19742. doi: 10.3934/math.20231005

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Abstract

The dynamical behavior of a 5-dimensional Lorenz model (5DLM) is investigated. Bifurcation diagrams address the chaotic and periodic behaviors associated with the bifurcation parameter. The Hamilton energy and its dependence on the stability of the dynamical system are presented. The global exponential attractive set (GEAS) is estimated in different 3-dimensional projection planes. A more conservative bound for the system is determined, that can be applied in synchronization and chaos control of dynamical systems. Finally, the finite time synchronization of the 5DLM, indicating the role of the ultimate bound for each variable, is studied. Simulations illustrate the effectiveness of the achieved theoretical results.

References

[1]	G. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcat. Chaos, 9 (1999), 1465–1466. https://doi.org/10.1142/S0218127499001024 doi: 10.1142/S0218127499001024
[2]	S. Celikovsky, G. Chen, On a generalized Lorenz canonical form of chaotic systems, Int. J. Bifurcat. Chaos, 12 (2002), 1789–1812. https://doi.org/10.1142/S0218127402005467 doi: 10.1142/S0218127402005467
[3]	E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130–141.
[4]	B. Saltzman, Finite amplitude free convection as an initial value problem, J. Atmos. Sci., 19 (1962), 329–341.
[5]	E. Lorenz, The predictability of hydrodynamic flow, Trans. N. Y. Acad. Sci., 25 (1963), 409–432.
[6]	X. Hu, B. Sang, N. Wang, The chaotic mechanisms in some jerk systems, AIMS Math., 7 (2022), 15714–15740. http://dx.doi.org/10.3934/math.2022861 doi: 10.3934/math.2022861
[7]	F. Zhang, K. Sun, Y. Chen, H. Zhang, C. Jiang, Parameters identification and adaptive tracking control of uncertain complex-variable chaotic systems with complex parameters, Nonlinear Dyn., 95 (2019), 3161–3176. https://doi.org/10.1007/s11071-018-04747-z doi: 10.1007/s11071-018-04747-z
[8]	S. Celikovsky, G. Chen, On the generalized Lorenz canonical form, Chaos Solit. Fract., 26 (2005), 1271–1276. https://doi.org/10.1016/j.chaos.2005.02.040 doi: 10.1016/j.chaos.2005.02.040
[9]	S. Celikovsky, G. Chen, Generalized Lorenz canonical form revisited, Int. J. Bifurcat. Chaos, 31 (2021), 2150079. https://doi.org/10.1142/S0218127421500796 doi: 10.1142/S0218127421500796
[10]	F. Zhang, S. Zhang, G. Chen, C. Li, Z. Li, C. Pan, Special attractors and dynamic transport of the hybrid-order complex Lorenz system, Chaos Solit. Fract., 164 (2022), 112700. https://doi.org/10.1016/j.chaos.2022.112700 doi: 10.1016/j.chaos.2022.112700
[11]	S. H. Salih, N. M. G. Al-Saidi, 3D-Chaotic discrete system of vector borne diseases using environment factor with deep analysis, AIMS Math., 7 (2022), 3972–3987. http://doi.org/10.3934/math.2022219 doi: 10.3934/math.2022219
[12]	A. Bushra Abdulshakoor M, W. Liu, Li-Yorke chaotic property of cookie-cutter systems, AIMS Math., 7 (2022), 13192–13207. https://doi.org/10.3934/math.2022727 doi: 10.3934/math.2022727
[13]	L. Chen, H. Yin, L. Yuan, A. M. Lopes, J. T. Machado, R. Wu, A novel color image encryption algorithm based on a fractional-order discrete chaotic neural network and DNA sequence operations, Front. Inform. Technol. Electron. Eng., 21 (2020), 866–879. https://doi.org/10.1631/FITEE.1900709 doi: 10.1631/FITEE.1900709
[14]	L. Chen, W. Pan, R. Wu, J. T. Machado, A. M. Lopes, Design and implementation of grid multi-scroll fractional-order chaotic attractors, Chaos, 26 (2016), 084303. https://doi.org/10.1063/1.4958717 doi: 10.1063/1.4958717
[15]	J. Li, N. Cui, A hyperchaos generated from Rabinovich system, AIMS Math., 8 (2023), 1410–1426. http://doi.org/10.3934/math.2023071 doi: 10.3934/math.2023071
[16]	B. W. Shen, Nonlinear feedback in a five-dimensional Lorenz model, J. Atmos. Sci., 71 (2014), 1701–1723. https://doi.org/10.1175/JAS-D-13-0223.1 doi: 10.1175/JAS-D-13-0223.1
[17]	S. Faghih-Naini, B. W. Shen, Quasi-periodic orbits in the five-dimensional non-dissipative Lorenz model: the role of the extended nonlinear feedback loop, Int. J. Bifurcat. Chaos, 28 (2018), 1850072. https://doi.org/10.1142/S0218127418500724 doi: 10.1142/S0218127418500724
[18]	B. W. Shen, Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term, Nonlin. Processes Geophys., 22 (2015), 749–764. https://doi.org/10.5194/npg-22-749-2015 doi: 10.5194/npg-22-749-2015
[19]	B. W. Shen, Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model, Nonlin. Processes Geophys., 23 (2016), 189–203. https://doi.org/10.5194/npg-23-189-2016 doi: 10.5194/npg-23-189-2016
[20]	B. W. Shen, Aggregated negative feedback in a generalized Lorenz model, Int. J. Bifurcat. Chaos, 29 (2019), 1950037. https://doi.org/10.1142/S0218127419500378 doi: 10.1142/S0218127419500378
[21]	G. Leonov, A. Bunin, N. Koksch, Attractor localization of the Lorenz system, ZAMM, 67 (1987), 649–656.
[22]	G. Leonov, N. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurc. Chaos Appl. Sci. Eng., 23 (2013), 1330002. https://doi.org/10.1142/S0218127413300024 doi: 10.1142/S0218127413300024
[23]	C. Feng, L. Li, Y. Liu, Z. Wei, Global dynamics of the chaotic disk dynamo system driven by noise, Complexity, 2020 (2020), 8375324. https://doi.org/10.1155/2020/8375324 doi: 10.1155/2020/8375324
[24]	Y. Li, Z. Wei, A. A. Aly, A 4D hyperchaotic Lorenz-type system: zero-Hopf bifurcation, ultimate bound estimation, and its variable-order fractional network, Eur. Phys. J. Spec. Top., 231 (2022), 1847–1858. https://doi.org/10.1140/epjs/s11734-022-00448-2 doi: 10.1140/epjs/s11734-022-00448-2
[25]	J. Jian, Z. Zhao, New estimations for ultimate boundary and synchronization control for a disk dynamo system, Nonlinear Anal. Hybrid Syst., 9 (2013), 56–66. https://doi.org/10.1016/j.nahs.2012.12.002 doi: 10.1016/j.nahs.2012.12.002
[26]	H. Saberi Nik, S. Effati, J. Saberi-Nadjafi, New ultimate bound sets and exponential finite-time synchronization for the complex Lorenz system, J. Complexity, 31 (2015), 715–730. https://doi.org/10.1016/j.jco.2015.03.001 doi: 10.1016/j.jco.2015.03.001
[27]	P. C. Rech, Hyperchaos and quasiperiodicity from a four-dimensional system based on the Lorenz system, Eur. Phys. J. B., 90 (2017), 251. https://doi.org/10.1140/epjb/e2017-80533-5 doi: 10.1140/epjb/e2017-80533-5
[28]	W. Gao, L. Yan, M-H. Saeedi, H. Saberi Nik, Ultimate bound estimation set and chaos synchronization for a financial risk system, Math. Comput. Simulat., 154 (2018), 19–33. https://doi.org/10.1016/j.matcom.2018.06.006 doi: 10.1016/j.matcom.2018.06.006
[29]	X. Zhang, Dynamics of a class of non-autonomous Lorenz-type systems, Int. J. Bifurcat. Chaos, 26 (2016), 1650208. https://doi.org/10.1142/S0218127416502084 doi: 10.1142/S0218127416502084
[30]	P. Swinnerton-Dyer, Bounds for trajectories of the Lorenz equations:an illustration of how to choose Liapunov functions, Phys. Lett A, 281 (2001), 161–167. https://doi.org/10.1016/S0375-9601(01)00109-8 doi: 10.1016/S0375-9601(01)00109-8
[31]	F. Chien, A. R. Chowdhury, H. Saberi Nik, Competitive modes and estimation of ultimate bound sets for a chaotic dynamical financial system, Nonlinear Dynam., 106 (2021), 3601–3614. https://doi.org/10.1007/s11071-021-06945-8 doi: 10.1007/s11071-021-06945-8
[32]	F. Chien, M. Inc, H-R.Yosefzade, H. Saberi Nik, Predicting the chaos and solution bounds in a complex dynamical system, Chaos Solitons Fract., 153 (2021), 111474. https://doi.org/10.1016/j.chaos.2021.111474 doi: 10.1016/j.chaos.2021.111474
[33]	H. Wang, X. Li, A note on "Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system" in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli, Appl. Math. Comput., 329 (2018), 1–4. https://doi.org/10.1016/j.amc.2018.01.027 doi: 10.1016/j.amc.2018.01.027
[34]	H. Wang, G. Dong, New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system, Appl. Math. Comput., 346 (2019), 272–286. https://doi.org/10.1016/j.amc.2018.10.006 doi: 10.1016/j.amc.2018.10.006
[35]	Y. Xie, P. Zhou, J. Ma, Energy balance and synchronization via inductive-coupling in functional neural circuits, Appl. Math. Model., 113 (2023), 175–187. https://doi.org/10.1016/j.apm.2022.09.015 doi: 10.1016/j.apm.2022.09.015
[36]	A. G. Radwan, K. Moaddy, K. N. Salama, S. Momani, I. Hashim, Control and switching synchronization of fractional order chaotic systems using active control technique, J. Adv. Res., 5 (2014), 125–132. https://doi.org/10.1016/j.jare.2013.01.003 doi: 10.1016/j.jare.2013.01.003
[37]	N. Cui, J. Li, A new 4D hyperchaotic system and its control, AIMS Math., 8 (2023), 905–923. http://dx.doi.org/10.3934/math.2023044 doi: 10.3934/math.2023044
[38]	Y. He, J. Peng, S. Zheng, Fractional-order financial system and fixed-time synchronization, Fractal Fract., 6 (2022), 507. https://doi.org/10.3390/fractalfract6090507 doi: 10.3390/fractalfract6090507
[39]	I. Ahmad, A. Ouannas, M. Shafiq, V. T. Pham, D. Baleanu, Finite-time stabilization of a perturbed chaotic finance model, J. Adv. Res., 32 (2021), 1–14. https://doi.org/10.1016/j.jare.2021.06.013 doi: 10.1016/j.jare.2021.06.013
[40]	D. Vivek, K. Kanagarajan, E. M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15, (2018), 15. https://doi.org/10.1007/s00009-017-1061-0 doi: 10.1007/s00009-017-1061-0
[41]	X. Leng, B. Du, S. Gu, S. He, Novel dynamical behaviors in fractional-order conservative hyperchaotic system and DSP implementation, Nonlinear Dynam., 109 (2022), 1167–1186. https://doi.org/10.1007/s11071-022-07498-0 doi: 10.1007/s11071-022-07498-0
[42]	A. M. A. El-Sayed, H. M. Nour, A. Elsaid, A. E. Matouk, A. Elsonbaty, Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system, Appl. Math. Model., 40 (2016), 3516–3534. https://doi.org/10.1016/j.apm.2015.10.010 doi: 10.1016/j.apm.2015.10.010
[43]	P. Zhou, X. K. Hu, Z. G. Zhu, J. Ma, What is the most suitable Lyapunov function? Chaos Solit. Fract., 150 (2021), 111154. https://doi.org/10.1016/j.chaos.2021.111154 doi: 10.1016/j.chaos.2021.111154

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