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Journal of Applied Nonlinear Dynamics 12(1) (2023) 125--131 | DOI:10.5890/JAND.2023.03.009

Mohammadreza Kheshti$^{1}$, Sajjad Taghvaei$^{1}$, Mohammad Salehi$^{1}$, Amir Masrour Baraijany$^{2}$

$^{1}$ Department of Mechanical Engineering, Shiraz University, Shiraz, Iran

$^{2}$ Department of Mechanical Engineering, Yazd University, Yazd, Iran

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Abstract

In this paper, the nonlinear behavior of an eleven-term 4-dimensional hyperchaotic Lorenz-type dynamic system is studied. The dynamic response of a hyperchaotic model is investigated. The phase portrait and Lyapunov exponent of the 4-D system are discussed. The phase portrait of the presented system shows the behavior which is like the Lorenz system's phase portrait. The nonlinear model is proved to be hyperchaotic since it has two positive Lyapunov exponents. The main goal of this research is to depict bifurcation and Poincare maps so as to investigate the occurrence of the periodic behavior, period-doubling, crisis, and chaotic motion. Therefore, bifurcation analysis is shown that by increasing the ${\theta }_{{2}}$, the behavior of the system will change from periodic into chaotic and vice versa. Also, the period-doubling, and crisis happened by changing the bifurcation parameter in a specific range. Finally, the Poincare map indicates that the chaotic motion appears when ${\theta }_{{2}}$ is 0.17.

References

1. [1]	Pecora, L.M. and Carroll, T.L. (1990), Synchronization in chaotic systems, Physical Review Letters, 64, 821-824.

2. [2]	Rong, C.G. and Ning, D.X. (1998), From Chaos to Order: Methodologies, Perspectives and Applications, 24, World Scientific.

3. [3]	Azar, A.T. and Vaidyanathan, S. (2014), Computational Intelligence Applications in Modeling and Control, 575, Springer.

4. [4]	Azar, A.T. and Vaidyanathan, S. (2015), Chaos Modeling and Control Systems Design, Springer.

5. [5]	Lorenz, E.N. (1963), Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20, 130-141.

6. [6]	Rössler, O.E. (1976), An equation for continuous chaos, Physics Letters A, 57, 397-398.

7. [7]	Conley, C. and Smoller, J.A. (1986), Bifurcation and stability of stationary solutions of the Fitz-Hugh-Nagumo equations.

8. [8]	Chen, G. and Ueta, T. (1999), Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9, 1465-1466.

9. [9]	Chen, H.K. and Lee, C.I. (2004), Anti-control of chaos in rigid body motion, Chaos, Solitons $\&$ Fractals, 21, 957-965.

10. [10]	Cai, G. and Tan, Z. (2007), Chaos synchronization of a new chaotic system via nonlinear control, Journal of Uncertain Systems, 1, 235-240.

11. [11]	Chua, L.O. (2007), Chua circuit, Scholarpedia, 2, 1488.

12. [12]	Arneodo, A., Coullet, P., and Tresser, C. (1981), Possible new strange attractors with spiral structure, Communications in Mathematical Physics, 79, 573-579.

13. [13]	Sprott, J.C. (1994), Some simple chaotic flows, Physical Review E, 50, R647.

14. [14]	Pham, V.T., Volos, C., Jafari, S., Wang, X., and Vaidyanathan, S. (2014), Hidden hyperchaotic attractor in a novel simple memristive neural network, Optoelectronics and Advanced Materials, Rapid Communications, 8, 1157-1163.

15. [15]	Pham, V.T., Volos, C., and Vaidyanathan, S. (2015), Chaotic attractor in a novel time-delayed system with a saturation function, in Handbook of Research on Advanced Intelligent Control Engineering and Automation, ed: IGI Global, pp. 230-258.

16. [16]	Pham, V.T., Vaidyanathan, S., Volos, C., Jafari, S., and Kingni, S.T. (2016), A no-equilibrium hyperchaotic system with a cubic nonlinear term, Optik-International Journal for Light and Electron Optics, 127, 3259-3265.

17. [17]	Kengne, J., Negou, A.N., and Njitacke, Z.T. (2017), Antimonotonicity, chaos and multiple attractors in a novel autonomous jerk circuit, International Journal of Bifurcation and Chaos, 27, 1750100.

18. [18]	Li, X. and Wu, R. (2014), Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system, Nonlinear Dynamics, 78, 279-288.

19. [19]	Chen, Y. and Yang, Q. (2014), Dynamics of a hyperchaotic Lorenz-type system, Nonlinear Dynamics, 77, 569-581.

20. [20]	Zarei, A. and Tavakoli, S. (2016), Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system, Applied Mathematics and Computation, 291, 323-339.

21. [21]	Chen, Y. (2017), The existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic system, Nonlinear Dynamics, 87, 1445-1452.

22. [22]	Tao, B., Xiao, M., Sun, Q., and Cao, J. (2018), Hopf bifurcation analysis of a delayed fractional-order genetic regulatory network model, Neurocomputing, 275, 677-686.

23. [23]	Wang, H. and Li, X. (2018), Hopf bifurcation and new singular orbits coined in a Lorenz-like system, Journal of Applied Analysis and Computation, 8, 1307-1325.

24. [24]	Al-Khedhairi, A., Elsonbaty, A., Abdel Kader, A.H., and Elsadany, A.A. (2019), Dynamic analysis and circuit implementation of a new 4D Lorenz-type hyperchaotic system, Mathematical Problems in Engineering, 2019.

25. [25]

    Vaidyanathan, S., Tlelo-Cuautle, E., Sambas, A., Dolvis, L.G., and Guillén-Fernández, O. (2020), A new four-dimensional two-scroll hyperchaos dynamical system with no rest point, bifurcation analysis, multi-stability, circuit simulation and FPGA design, International Journal of Computer Applications in Technology, 63, 147-159.

26. [26]	Wang, H. and Zhang, F. (2020), Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system, Discrete $\&$ Continuous Dynamical Systems-Series B, 25(5), 1791.

27. [27]	Vaidyanathan, S. (2014), Qualitative analysis and control of an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities, International Journal of Control Theory and Applications, 7, 35-47.

28. [28]	Vaidyanathan, S. and Sampath, S. (2016), Complete synchronization of hyperchaotic systems via novel sliding mode control, in Advances in Chaos Theory and Intelligent Control, ed: Springer, pp. 327-347.