Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system
Authors
T. I. Rasul
- Department of Mathematics, Faculty of Science, Soran University, Soran, Kurdistan Region, Iraq.
R. H. Salih
- Department of Mathematics, College of Basic Education, University of Raparin, Rania, Kurdistan Region, Iraq.
Abstract
This paper is devoted to investigating the local bifurcation of a chaotic jerk system. The local stability of equilibrium points is analyzed, as well as the existence of transcritical bifurcation at the origin. For the proposed jerk system, the Hopf and Zero-Hopf bifurcations are investigated at the origin. Moreover, a zero-Hopf equilibrium point at the origin is characterized for the system. By using the averaging theory of first order, a limit cycle can be bifurcated from the zero-Hopf equilibrium located at the origin. Liapunov quantities techniques are used to investigate the cyclicity of the system. It is shown that three limit cycles can be bifurcated from the origin. Finally, both self-excited chaotic attractors and hidden chaotic attractors are studied for special cases of the chaotic jerk systems using bifurcation diagrams, Lyapunov exponents, and cross-sections. All results reported in this study have been obtained using Maple software.
Share and Cite
ISRP Style
T. I. Rasul, R. H. Salih, Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system, Journal of Mathematics and Computer Science, 35 (2024), no. 3, 319--335
AMA Style
Rasul T. I., Salih R. H., Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system. J Math Comput SCI-JM. (2024); 35(3):319--335
Chicago/Turabian Style
Rasul, T. I., Salih, R. H.. "Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system." Journal of Mathematics and Computer Science, 35, no. 3 (2024): 319--335
Keywords
- Transcritical bifurcation
- zero hopf
- Hopf bifurcation
- jerk system
- self-excited attractors
- hidden attractors
MSC
References
-
[1]
T. Bonny, S. Vaidyanathan, A. Sambas, K. Benkouide, W. A. Nassan, O. Naqaweh, Multistability and bifurcation analysis of a novel 3d jerk system: Electronic circuit design, fpga implementation, and image cryptography scheme, IEEE Access, 11 (2023), 1–17
-
[2]
F. Braun, A. C. Mereu, Zero-Hopf bifurcation in a 3D jerk system, Nonlinear Anal. Real World Appl., 59 (2021), 8 pages
-
[3]
S. Cang, Y. Li, R. Zhang, Z. Wang, Hidden and self-excited coexisting attractors in a Lorenz-like system with two equilibrium points, Nonlinear Dyn., 95 (2019), 381–390
-
[4]
Q. Chen, B. Li, W. Yin, X. Jiang, X. Chen, Bifurcation, chaos and fixed-time synchronization of memristor cellular neural networks, Chaos Solitons Fractals, 171 (2023), 10 pages
-
[5]
M.-F. Danca, Hidden chaotic attractors in fractional-order systems, Nonlinear Dynam., 89 (2017), 577–586
-
[6]
Z. Diab, J. L. G. Guirao, J. A. Vera, Zero-Hopf bifurcation in a generalized Genesio differential equation, Mathematics, 9 (2021), 1–11
-
[7]
J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York (2013)
-
[8]
B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and applications of Hopf bifurcation, Cambridge University Press, Cambridge-New York (1981)
-
[9]
X. Hu, B. Sang, N. Wang, The chaotic mechanisms in some jerk systems, AIMS Math., 7 (2022), 15714–15740
-
[10]
S. Jafari, J. C. Sprott, Simple chaotic flows with a line equilibrium, Chaos Solitons Fractals, 57 (2013), 79–84
-
[11]
M. Joshi, A . Ranjan, An autonomous simple chaotic jerk system with stable and unstable equilibria using reverse sine hyperbolic functions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 10 pages
-
[12]
L. K. Kengne, H. T. Kamdem Tagne, J. R. Mboupda Pone, J. Kengne, Dynamics, control and symmetry-breaking aspects of a new chaotic Jerk system and its circuit implementation, Eur. Phys. J. Plus, 135 (2020),
-
[13]
J. Kengne, Z. T. Njitacke, H. B. Fotsin, Dynamical analysis of a simple autonomous jerk system with multiple attractors, Nonlinear Dynam., 83 (2016), 751–765
-
[14]
J. Kengne, V. R. F. Signing, J. C. Chedjou, G. D. Leutcho, Nonlinear behavior of a novel chaotic jerk system: antimonotonicity, crises, and multiple coexisting attractors, Int. J. Dyn. Control, 6 (2018), 468–485
-
[15]
N. V. Kuznetsov, Hidden attractors in fundamental problems and engineering models: A short survey, Lecture Notes in Electrical Engineering, Springer, Cham, (2016), 13–25
-
[16]
N. V. Kuznetsov, G. A. Leonov, V. I. Vagaitsev, Analytical-numerical method for attractor localization of generalized Chua’s system, IFAC Proc. Vol., 43 (2010), 29–33
-
[17]
C. LĖazureanu, J. Cho, On Hopf and fold bifurcations of jerk systems, Mathematics, 11 (2023), 1–15
-
[18]
G. A. Leonov, N. V. Kuznetsov, V. I. Vagaitsev, Localization of hidden Chua’s attractors, Phys. Lett. A, 375 (2011), 2230–2233
-
[19]
J. Li, Y. Liu, Z. Wei, Zero-Hopf bifurcation and Hopf bifurcation for smooth Chua’s system, Adv. Difference Equ., 2018 (2018), 17 pages
-
[20]
M. Molaie, S. Jafari, J. C. Sprott , S. M. R. H. Golpayegani, Simple chaotic flows with one stable equilibrium, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 7 pages
-
[21]
L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York (2013)
-
[22]
K. Rajagopal, S. T. Kingni, G. F. Kuiate, V. K. Tamba, V.-T. Pham, Autonomous jerk oscillator with cosine hyperbolic nonlinearity: analysis, FPGA implementation, and synchronization, Adv. Math. Phys., 2018 (2018), 12 pages
-
[23]
R. H. Salih, Hopf bifurcation and centre bifurcation in three dimensional Lotka-Volterra systems, PhD thesis, Plymouth University, (2015)
-
[24]
R. H. Salih, M. S. Hasso, S. H. Ibrahim, Centre Bifurcations for a Three-Dimensional System with Quadratic Terms, Zanco J. Pure Appl. Sci., 32 (2020), 62–71
-
[25]
R. H. Salih, B. M. Mohammed, Hopf bifurcation analysis of a Chaotic system, Zanco J. Pure Appl. Sci., 34 (2022), 87–100
-
[26]
B. Sang, B. Huang, Bautin bifurcations of a financial system, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 22 pages
-
[27]
B. Sang, B. Huang, Zero-Hopf bifurcations of 3D jerk quadratic system, Mathematics, 8 (2020), 1–17
-
[28]
A. Singh, V. S. Sharma, Codimension-2 bifurcation in a discrete predator–prey system with constant yield predator harvesting, Int. J. Biomath., 16 (2023), 27 pages
-
[29]
J. C. Sprott, Elegant chaos: algebraically simple chaotic flows, World Scientific Publishing Co., Singapore (2010)
-
[30]
S. Vaidyanathan, S. T. Kingni, A. Sambas, M. A. Mohamed, M. Mamat, A new chaotic jerk system with three nonlinearities and synchronization via adaptive backstepping control, Int. J. Eng. Technol., 7 (2018), 1936–1943
-
[31]
F. Verhulst, Nonlinear differential equations and dynamical systems, Springer-Verlag, Berlin (2006)
-
[32]
H. Wang, G. Ke, J. Pan, Q. Su, Modeling, dynamical analysis and numerical simulation of a new 3D cubic Lorenz-like system, Sci. Rep., 13 (2023), 1–15
-
[33]
X. Wang, V.-T. Pham, S. Jafari, C. Volos, J. M. Munoz-Pacheco, E. Tlelo-Cuautle, A new chaotic system with stable equilibrium: From theoretical model to circuit implementation, IEEE Access, 5 (2017), 8851–8858
-
[34]
C. Wannaboon, T. Masayoshi, An autonomous chaotic oscillator based on hyperbolic tangent nonlinearity, In: 2015 15th International Symposium on Communications and Information Technologies (ISCIT), IEEE Access, (2015), 323–326
-
[35]
Z. Wei, Dynamical behaviors of a chaotic system with no equilibria, Phys. Lett. A, 376 (2011), 102–108
-
[36]
A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D, 16 (1985), 285–317
-
[37]
Q. Yang, L. Yang, B. Ou, Hidden hyperchaotic attractors in a new 5D system based on chaotic system with two stable node-foci, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 21 pages