A new 5D hyperjerk system: hyperchaos, Hopf bifurcation, synchronization, circuit design and an encryption application

Junhong Li; Ning Cui; Liang Wang; Junhong Li; Ning Cui; Liang Wang

doi:10.3934/era.2026303

Electronic Research Archive

2026, Volume 34, Issue 9: 6999-7026. doi: 10.3934/era.2026303

Previous Article Next Article

Research article Special Issues

A new 5D hyperjerk system: hyperchaos, Hopf bifurcation, synchronization, circuit design and an encryption application

1.
School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong 521041, China
2.
Coordinated Innovation Center for Computable Modelling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China

Received: 05 June 2026 Revised: 20 July 2026 Accepted: 24 July 2026 Published: 14 August 2026

Different from the classical triangular-structure hyperjerk forms, this work constructed a new five-dimensional (5D) hyperjerk system from a novel perspective and analyzed its dissipativity as well as the stability of the equilibrium point. By employing methods such as Lyapunov exponents, bifurcation diagrams, the 0-1 test, and Poincaré sections, the existence of a hyperchaotic attractor and the rich dynamical behaviors of the system were revealed. Based on normal form theory and symbolic computations, the existence of Hopf bifurcations and the stability of the bifurcation solutions were further investigated, and numerical examples were provided to validate the theoretical findings. By incorporating Lyapunov functions, we designed a suitable nonlinear controller that achieves global asymptotic synchronization of the hyperchaotic system. Simulation results demonstrated that the drive-response system achieves synchronization rapidly. In addition, an electronic circuit design scheme for the 5D hyperchaotic system was presented. The circuit simulation results were in good agreement with the numerical simulations, thereby verifying the physical realizability of the system. Finally, the practical application of the proposed hyperchaotic system in information encryption and decryption was explored. Based on this 5D hyperchaotic hyperjerk system, we developed encryption and decryption algorithms. From the analysis results of multiple metrics, including the correlation coefficient, information entropy, key space, number of pixel change rate, and unified average changing intensity, the proposed algorithm demonstrated favorable confidentiality and robust security performance.
- hyperchaotic hyperjerk system,
- Hopf bifurcation,
- hyperchaos synchronization,
- hyperchaotic circuit,
- hyperchaos encryption
Citation: Junhong Li, Ning Cui, Liang Wang. A new 5D hyperjerk system: hyperchaos, Hopf bifurcation, synchronization, circuit design and an encryption application[J]. Electronic Research Archive, 2026, 34(9): 6999-7026. doi: 10.3934/era.2026303

Related Papers:

Abstract

Different from the classical triangular-structure hyperjerk forms, this work constructed a new five-dimensional (5D) hyperjerk system from a novel perspective and analyzed its dissipativity as well as the stability of the equilibrium point. By employing methods such as Lyapunov exponents, bifurcation diagrams, the 0-1 test, and Poincaré sections, the existence of a hyperchaotic attractor and the rich dynamical behaviors of the system were revealed. Based on normal form theory and symbolic computations, the existence of Hopf bifurcations and the stability of the bifurcation solutions were further investigated, and numerical examples were provided to validate the theoretical findings. By incorporating Lyapunov functions, we designed a suitable nonlinear controller that achieves global asymptotic synchronization of the hyperchaotic system. Simulation results demonstrated that the drive-response system achieves synchronization rapidly. In addition, an electronic circuit design scheme for the 5D hyperchaotic system was presented. The circuit simulation results were in good agreement with the numerical simulations, thereby verifying the physical realizability of the system. Finally, the practical application of the proposed hyperchaotic system in information encryption and decryption was explored. Based on this 5D hyperchaotic hyperjerk system, we developed encryption and decryption algorithms. From the analysis results of multiple metrics, including the correlation coefficient, information entropy, key space, number of pixel change rate, and unified average changing intensity, the proposed algorithm demonstrated favorable confidentiality and robust security performance.

References

[1]	M. Alawida, A novel image encryption algorithm based on cyclic chaotic map in industrial iot environments, IEEE Trans. Ind. Inf., 20 (2024), 10530–10541. https://doi.org/10.1109/TII.2024.3395631 doi: 10.1109/TII.2024.3395631
[2]	V. Balasubramanian, J. M. Magan, Q. Wu, Quantum chaos, integrability, and late times in the Krylov basis, Phys. Rev. E, 111 (2025), 014218. https://doi.org/10.1103/PhysRevE.111.014218 doi: 10.1103/PhysRevE.111.014218
[3]	M. Tian, S. Yan, Y. Uyaroğlu, A. Mohammadzadeh, H. Taghavifar, C. Zhang, Type-3 fuzzy discrete-time modeling and synchronization of financial chaotic/hyperchaotic systems, Nonlinear Dyn., 113 (2025), 28427–28447. https://doi.org/10.1007/s11071-025-11533-1 doi: 10.1007/s11071-025-11533-1
[4]	E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130–141. https://doi.org/10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2
[5]	O. E. Rossler, An equation for hyperchaos, Phys. Lett. A, 71 (1979), 155–157. https://doi.org/10.1016/0375-9601(79)90150-6
[6]	J. C. Sprott, Some simple chaotic jerk functions, Am. J. Phys., 65 (1997), 537–543. https://doi.org/10.1119/1.18585 doi: 10.1119/1.18585
[7]	G. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation Chaos, 9 (1999), 1465–1466. https://doi.org/10.1142/S0218127499001024 doi: 10.1142/S0218127499001024
[8]	J. Lü, G. Chen, A new chaotic attractor coined, Int. J. Bifurcation Chaos, 12 (2002), 659–661. https://doi.org/10.1142/S0218127402004620 doi: 10.1142/S0218127402004620
[9]	J. Chen, W. Lv, L. Y. Zhang, Z. Hua, L. Zhang, Z. Zhu, Power-$\alpha$ chaotic system with robust chaos, Nonlinear Dyn., 113 (2025), 12185–12197. https://doi.org/10.1007/s11071-024-10680-1 doi: 10.1007/s11071-024-10680-1
[10]	S. Yan, Y. Cui, X. Sun, A jerk chaotic system with bistable locally active memristor and its analysis of multi-scroll formation mechanism, Eur. Phys. J. Plus, 139 (2024), 30. https://doi.org/10.1140/epjp/s13360-023-04829-x doi: 10.1140/epjp/s13360-023-04829-x
[11]	S. Wang, J. He, Design of new chaotic system with multi-scroll attractor by using variable transformation and its application, Phys. Scr., 100 (2025), 025230. https://doi.org/10.1088/1402-4896/adaa39 doi: 10.1088/1402-4896/adaa39
[12]	R. Lan, Y. Wang, T. Xia, L. Lin, Dynamical analysis and FPGA implementation of a memristive non-Hamiltonian conservative hyperchaotic system with extreme multistability, Integration, 106 (2026), 102565. https://doi.org/10.1016/j.vlsi.2025.102565 doi: 10.1016/j.vlsi.2025.102565
[13]	Q. Yang, W. M. Osman, C. Chen, A new 6D hyperchaotic system with four positive Lyapunov exponents coined, Int. J. Bifurcation Chaos, 25 (2015), 1550060. https://doi.org/10.1142/S0218127415500601 doi: 10.1142/S0218127415500601
[14]	K. H. Moussa, A novel hyperchaotic four-dimensional memristive log-logistic sine map with histogram equalisation for image encryption, Nonlinear Dyn., 113 (2025), 31725–31754. https://doi.org/10.1007/s11071-025-11694-z doi: 10.1007/s11071-025-11694-z
[15]	K. Aldwoah, E. I. Hassan, M. Alsharafi, A. F. Alharbi, Hyperchaotic system for secure communication: A modified 4D model and its dynamics, J. Nonlinear Math. Phys., 32 (2025), 90. https://doi.org/10.1007/s44198-025-00348-8 doi: 10.1007/s44198-025-00348-8
[16]	C. Zhao, H. Ren, Image encryption based on infinite dimensional hyper-chaotic multi-attractors, Nonlinear Dyn., 100 (2020), 679–698. https://doi.org/10.1007/s11071-020-05526-5 doi: 10.1007/s11071-020-05526-5
[17]	K. A. Abed, S. F. Al-Azzawi, O. S. Qasim, Electronic circuit and image encryption using a novel simple 4D hyperchaotic system, Phys. Scr., 100 (2025), 015210. https://doi.org/10.1088/1402-4896/ad941d doi: 10.1088/1402-4896/ad941d
[18]	K. E. Chlouverakis, J. C. Sprott, Chaotic hyperjerk systems, Chaos, Solitons Fractals, 28 (2006), 739–746. https://doi.org/10.1016/j.chaos.2005.08.019
[19]	F. Y. Dalkiran, J. C. Sprott, Simple chaotic hyperjerk systems, Int. J. Bifurcation Chaos, 26 (2016), 1650189. https://doi.org/10.1142/S0218127416501893 doi: 10.1142/S0218127416501893
[20]	S. Vaidyanathan, F. Hannachi, M. A. Mohamed, A. Sambas, C. Aruna, R. Ramesh, A new chaotic hyperjerk system with a half-line of equilibrium points, its dynamic analysis, multistability, circuit simulation and anti-synchronization via backstepping control, Arch. Control Sci., 35 (2025), 123–143. https://doi.org/10.24425/acs.2025.153960 doi: 10.24425/acs.2025.153960
[21]	K. Rajagopal, Y. Shekofteh, F. Nazarimehr, C. Li, S. Jafari, A new chaotic multi-stable hyperjerk system with various types of attractors, Indian J. Phys., 96 (2022), 1501–1507. https://doi.org/10.1007/s12648-021-02075-4 doi: 10.1007/s12648-021-02075-4
[22]	L. Jiang, J. Li, W. Zhang, Bifurcations and chaos dynamics of a hyperjerk system with antimonotonicity, Eur. Phys. J. Plus, 135 (2020), 767. https://doi.org/10.1140/epjp/s13360-020-00786-x doi: 10.1140/epjp/s13360-020-00786-x
[23]	E. Zambrano-Serrano, A. Anzo-Hernández, A novel antimonotic hyperjerk system: Analysis, synchronization and circuit design, Phys. D, 424 (2021), 132927. https://doi.org/10.1016/j.physd.2021.132927 doi: 10.1016/j.physd.2021.132927
[24]	G. D. Leutcho, J. Kengne, L. K. Kengne, A. Akgul, V. Pham, S. Jafari, A novel chaotic hyperjerk circuit with bubbles of bifurcation: Mixed-mode bursting oscillations, multistability, and circuit realization, Phys. Scr., 95 (2020), 075216. https://doi.org/10.1088/1402-4896/ab92da doi: 10.1088/1402-4896/ab92da
[25]	R. Y. Taha, N. H. Hussein, A. I. Amen, Hopf and zero-double Hopf bifurcations of 5D hyperjerk oscillator network system, Int. J. Bifurcation Chaos, 35 (2025), 2550029. https://doi.org/10.1142/S0218127425500294 doi: 10.1142/S0218127425500294
[26]	B. Bao, M. A. Peol, H. Bao, M. Chen, H. Li, B. Chen, No-argument memristive hyper-jerk system and its coexisting chaotic bubbles boosted by initial conditions, Chaos, Solitons Fractals, 144 (2021), 110744. https://doi.org/10.1016/j.chaos.2021.110744 doi: 10.1016/j.chaos.2021.110744
[27]	B. Zhang, J. Wang, Y. Feng, Z. Zhang, Finite-time projective synchronization of hyperjerk systems modeled with fuzzy recurrent neural networks, IEEE Trans. Fuzzy Syst., 32 (2024), 4482–4495. https://doi.org/10.1109/TFUZZ.2024.3401112 doi: 10.1109/TFUZZ.2024.3401112
[28]	S. Vaidyanathan, A. Akgul, S. Kaçar, U. Çavuşoğlu, A new 4-D chaotic hyperjerk system, its synchronization, circuit design and applications in RNG, image encryption and chaos-based steganography, Eur. Phys. J. Plus, 133 (2018), 46. https://doi.org/10.1140/epjp/i2018-11872-8 doi: 10.1140/epjp/i2018-11872-8
[29]	X. Wang, S. Vaidyanathan, C. Volos, V. Pham, T. Kapitaniak, Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors, Nonlinear Dyn., 89 (2017), 1673–1687. https://doi.org/10.1007/s11071-017-3542-x doi: 10.1007/s11071-017-3542-x
[30]	Q. Yang, C. Chen, A 5D hyperchaotic system with three positive Lyapunov exponents coined, Int. J. Bifurcation Chaos, 23 (2013), 1350109. https://doi.org/10.1142/s0218127413501095 doi: 10.1142/s0218127413501095
[31]	S. Iqbal, J. Wang, Analysis of a novel fractional order hyper-chaotic system: Dynamics, stability and synchronization analysis, Phys. Lett. A, 555 (2025), 130770. https://doi.org/10.1016/j.physleta.2025.130770 doi: 10.1016/j.physleta.2025.130770
[32]	A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D, 16 (1985), 285–317. https://doi.org/10.1016/0167-2789(85)90011-9 doi: 10.1016/0167-2789(85)90011-9
[33]	G. A. Gottwald, I. Melbourne, A new test for chaos in deterministic systems, Proc. R. Soc. A., 460 (2004), 603–611. https://doi.org/10.1098/rspa.2003.1183 doi: 10.1098/rspa.2003.1183
[34]	J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, $1^{st}$ edition, Springer, 1983. https://doi.org/10.1007/978-1-4612-1140-2
[35]	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, $2^nd$ edition, Springer, 1998.
[36]	D. Biswas, T. Banerjee, A simple chaotic and hyperchaotic time-delay system: Design and electronic circuit implementation, Nonlinear Dyn., 83 (2016), 2331–2347. https://doi.org/10.1007/s11071-015-2484-4 doi: 10.1007/s11071-015-2484-4
[37]	X. Cai, C. Liu, Y. Wang, H. Zhang, A novel 4D chaotic system with nonhyperbolic hyperbolic shape equilibrium points: analysis, circuit implementation and color image encryption, Int. J. Mod. Phys. B, 33 (2019), 1950383. https://doi.org/10.1142/S0217979219503831 doi: 10.1142/S0217979219503831
[38]	J. Zhang, E. Liu, Circuit design and image encryption of CNN chaotic system based on memristor, Eur. Phys. J. B, 97 (2024), 100. https://doi.org/10.1140/epjb/s10051-024-00743-y doi: 10.1140/epjb/s10051-024-00743-y
[39]	Y. Wang, Z. Liu, J. Ma, H. He, A pseudorandom number generator based on piecewise logistic map, Nonlinear Dyn., 83 (2016), 2373–2391. https://doi.org/10.1007/s11071-015-2488-0 doi: 10.1007/s11071-015-2488-0
[40]	Y. Wang, Z. Zhang, G. Wang, D. Liu, A pseudorandom number generator based on a 4D piecewise logistic map with coupled parameters, Int. J. Bifurcation Chaos, 29 (2019), 1950124. https://doi.org/10.1142/s0218127419501244 doi: 10.1142/s0218127419501244
[41]	S. Yan, L. Li, B. Gu, Xi Sun, Y. Ren, Y. Zhang, A color image encryption scheme based on chaotic mapping, chaotic system, and DNA coding, Appl. Intell., 53 (2023), 31181–31206. https://doi.org/10.1007/s10489-023-04759-2 doi: 10.1007/s10489-023-04759-2
[42]	K. M. Hosny, S. T. Kamal, M. M. Darwish, A novel color image encryption based on fractional shifted Gegenbauer moments and 2D logistic-sine map, Visual Comput., 39 (2023), 1027–1044. https://doi.org/10.1007/s00371-021-02382-1 doi: 10.1007/s00371-021-02382-1

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)