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Department of Mathematics, College of Education, Salahaddin University-Erbil, Erbil, Iraq

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Niazy Hady Hussein

https://orcid.org/0000-0003-1687-3888

Department of Mathematics, College of Education, Salahaddin University-Erbil, Erbil, Iraq

E-mail Address: niazy.hussein@soran.edu.iq

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Azad Ibrahim Amen

https://orcid.org/0000-0002-9310-6474

Department of Mathematics, College of Basic Education, Salahaddin University-Erbil, Erbil, Iraq

E-mail Address: azad.amen@su.edu.krd

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https://doi.org/10.1142/S0218127425500294Cited by:0 (Source: Crossref)

Abstract

In this paper, we consider a five-dimensional dynamic hyperjerk system. This kind of system illustrates the use of chaos induction for lightweight picture encryption. Significant advances in internet and multimedia technologies have facilitated the seamless exchange and transmission of confidential information, particularly in the context of images. Here, the local bifurcation of the system is examined, with explicit conditions provided to determine when the zero and zero-double Hopf equilibrium points undergo changes in this particular type of system. Furthermore, the existence of periodic solutions that bifurcate from these types of equilibrium points for the five-dimensional network hyperjerk system is studied. Consequently, a periodic solution of the system is obtained, and its stability is classified. This periodic solution bifurcates from the zero-double Hopf bifurcation equilibrium point, and the first-order averaging method is employed to determine it. Furthermore, two periodic solutions are obtained, and their stability is classified for the system from the projection approach. Notably, these two periodic solutions could not be identified using the averaging method of the first order.

Keywords:

References

Al-Raeei, M. [2021] “Applying fractional quantum mechanics to systems with electrical screening effects,” Chaos Solit. Fract. 150, 111209. Crossref, Web of Science, Google Scholar
Amen, A. I. & Salih, R. H. [2008] “Limit cycles of Lorenz system with Hopf bifurcation,” Al-Rafiden J. Comput. Sci. Math. 5, 81–99. Google Scholar
Baldomá, I. & Seara, T. M. [2006] “Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity,” J. Nonlin. Sci. 16, 543–582. Crossref, Web of Science, Google Scholar
Baldoma, I. & Seara, T. M. [2008] “The inner equation for generic analytic unfoldings of the Hopf-zero singularity,” Discrete Contin. Dyn. Syst. Ser. B 10, 323–347. Crossref, Web of Science, Google Scholar
Berrehail, C. E. & Bouslah, Z. [2022] “Periodic solutions for a class of fifth-order differential equations,” Arab J. Math. Sci. 1, 2–13. Google Scholar
Berrehail, C. E. C. E. & Makhlouf, A. [2022] “Periodic solutions for a class of perturbed fifth-order autonomous differential equations via averaging theory,” Int. J. Nonlin. Anal. Appl. 13, 2479–2491. Web of Science, Google Scholar
Broer, H. & Vegter, G. [1984] “Subordinate šil’nikov bifurcations near some singularities of vector fields having low codimension,” Ergod. Theory Dyn. Syst. 4, 509–525. Crossref, Google Scholar
Buică, A. & Llibre, J. [2004] “Averaging methods for finding periodic orbits via brouwer degree,” Bull. Sci. Math. 128, 7–22. Crossref, Web of Science, Google Scholar
Buica, A., Françoise, J.-P. & Llibre, J. [2007] “Periodic solutions of nonlinear periodic differential systems with a small parameter,” Commun. Pure Appl. Anal. 6, 103. Crossref, Web of Science, Google Scholar
Champneys, A. & Kirk, V. [2004] “The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities,” Phys. D: Nonlin. Phenom. 195, 77–105. Crossref, Web of Science, Google Scholar
Chen, Z. [2022] “Complex bursting patterns and their bifurcation mechanisms in the $ϕ$ $ϕ$ 6-van der Pol system with external and parametric excitations,” Pramana 96, 157. Crossref, Google Scholar
Glaubitz, A. & Fu, F. [2023] “Population heterogeneity in vaccine coverage impacts epidemic thresholds and bifurcation dynamics,” Heliyon 9, e19094. Crossref, Web of Science, Google Scholar
Guckenheimer, J. [1981] On a Codimension Two Bifurcation (Springer). Crossref, Google Scholar
Guckenheimer, J. & Holmes, P. [2013] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer Science & Business Media). Google Scholar
Hassard, B. D., Kazarinoff, N. D. & Wan, Y.-H. [1981] Theory and Applications of Hopf Bifurcation (CUP Archive). Google Scholar
Husien, A. M. & Amen, A. I. [2024] “Hopf and zero-Hopf bifurcation analysis for a chaotic system,” Int. J. Bifurcation and Chaos 34, 2450104. Link, Web of Science, Google Scholar
Khan, J. S. & Ahmad, J. [2019] “Chaos based efficient selective image encryption,” Multidimens. Syst. Signal Process. 30, 943–961. Crossref, Web of Science, Google Scholar
Kuznetsov, Y. A., Kuznetsov, I. A. & Kuznetsov, Y. [1998] Elements of Applied Bifurcation Theory (Springer). Google Scholar
Kuznetsov, Y. A. & Kuznetsov, Y. A. [2004] Numerical Analysis of Bifurcations (Springer). Crossref, Google Scholar
Li, S., Chen, G., Cheung, A., Bhargava, B. & Lo, K.-T. [2007] “On the design of perceptual MPEG-video encryption algorithms,” IEEE Trans. Circuits Syst. Video Technol. 17, 214–223. Crossref, Web of Science, Google Scholar
Li, L., Abd-El-Atty, B., Abd El-Latif, A. A. & Ghoneim, A. [2017] “Quantum color image encryption based on multiple discrete chaotic systems,” 2017 Federated Conf. on Computer Science and Information Systems (FedCSIS) (IEEE), pp. 555–559. Crossref, Google Scholar
Llibre, J., Mereu, A. C. & Teixeira, M. A. [2010] “Limit cycles of the generalized polynomial Liénard differential equations,” Math. Proc. Camb. Philos. Soc. 148, 363–383. Crossref, Web of Science, Google Scholar
Luo, Y., Zhou, R., Liu, J., Cao, Y. & Ding, X. [2018] “A parallel image encryption algorithm based on the piecewise linear chaotic map and hyper-chaotic map,” Nonlin. Dyn. 93, 1165–1181. Crossref, Web of Science, Google Scholar
Maoan, H. [1998] “Existence of periodic orbits and invariant tori in codimension two bifurcations of three dimensional systems,” J. Syst. Sci. Math. Sci. 18, 403. Google Scholar
Nestor, T., De Dieu, N. J., Jacques, K., Yves, E. J., Iliyasu, A. M. & Abd El-Latif, A. A. [2019] “A multidimensional hyperjerk oscillator: Dynamics analysis, analogue and embedded systems implementation, and its application as a cryptosystem,” Sensors 20, 83. Crossref, Web of Science, Google Scholar
Nkapkop, J. D. D., Effa, J. Y., Borda, M. & Terebes, R. [2015] “A novel fast and secure chaos-based algorithm for image encryption,” Innovative Security Solutions for Information Technology and Communications: 8th Int. Conf., SECITC 2015, Bucharest, Romania, June 11–12, 2015, Revised Selected Papers 8 (Springer), pp. 87–101. Crossref, Google Scholar
Rabia, A. A. & Makhlouf, A. [2022] “Periodic solutions for differential systems in $ℝ^{5}$ and $ℝ^{6}$ ,” J. Dyn. Control Syst. 29, 647–657. Crossref, Google Scholar
Scheurle, J. & Marsden, J. [1984] “Bifurcation to quasi-periodic tori in the interaction of steady state and Hopf bifurcations,” SIAM J. Math. Anal. 15, 1055–1074. Crossref, Web of Science, Google Scholar
Sellami, N., Mellal, R., Cherif, B. B. & Idris, S. A. [2021] “On the limit cycles for a class of perturbed fifth-order autonomous differential equations,” Discrete Dyn. Nat. Soc. 2021, 6996805. Crossref, Web of Science, Google Scholar
Sotomayor, J., Mello, L. F. & Braga, D. D. C. [2007] “Bifurcation analysis of the Watt governor system,” Comput. Appl. Math. 26, 19–44. Crossref, Web of Science, Google Scholar
Sotomayor, J., Mello, L. F. & Braga, D. D. C. [2008] “Hopf bifurcations in a Watt governor with a spring,” J. Nonlin. Math. Phys. 15, 288–299. Crossref, Web of Science, Google Scholar
Tabet, A. E. & Makhlouf, A. [2023] “Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory,” Nonauton. Dyn. Syst. 10, 20230104. Crossref, Google Scholar
Taha, R. Y., Amen, A. I. & Hussein, N. H. [2023] “Darboux and analytic integrability of five-dimensional semiclassical Jaynes-Cummings system,” Math. Methods Appl. Sci. 46, 13289–13303. Crossref, Web of Science, Google Scholar
Tsafack, N., Kengne, J., Abd-El-Atty, B., Iliyasu, A. M., Hirota, K. & Abd EL-Latif, A. A. [2020] “Design and implementation of a simple dynamical 4-d chaotic circuit with applications in image encryption,” Inf. Sci. 515, 191–217. Crossref, Web of Science, Google Scholar
Yang, Q. & Chen, C. [2013] “A 5d hyperchaotic system with three positive Lyapunov exponents coined,” Int. J. Bifurcation and Chaos 23, 1350109. Link, Web of Science, Google Scholar