CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM

Michael Kopp

CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM

Abstract

This paper introduces a new four-dimensional (4-D) dynamical system composed of only seven terms: four linear terms, one nonlinear term involving the hyperbolic tangent function, one absolute value function term, and a constant. The new 4-D system does not have any equilibrium points and is capable of producing hidden attractors. The paper includes a detailed dynamical analysis, which encompasses bifurcation diagrams, Lyapunov exponents, Kaplan-Yorke dimensions, and bias amplification. Additionally, the theoretical model is verified through an electronic simulation of the system using Multisim© 14.2. The paper also demonstrates the synchronization of two identical 4-D hyperchaotic systems using the active control method. The proposed simple dynamic system exhibits a rather complex chaotic behavior and may find applications in various practical domains.

Keywords

Thanks

I thank anonymous reviewer for valuable suggestions and comments.

References

Mengue, A. D., Essebe, D. E., Essimbi, B. Z., (2024), High-dimensional hyperchaos and its control in a modified laser system subjected to optical injection, Opt. Quant. Electron., 56, pp. 1101.
Hunaish, A. S., Tahir, F. R., Abbood, H. A., (2021), Hyperchaos from DTC Induction Motor Drive System, IFAC-PapersOnLine, 54, pp. 81-86.
Shvets, A., Sirenko, V., (2019), Hyperchaos in Oscillating Systems with Limited Excitation, Springer Proceedings in Complexity, Springer, Cham, pp. 265-273.
Lin, H., Wang, C., Cui, L., et al., (2022), Hyperchaotic memristive ring neural network and application in medical image encryption, Nonlinear Dyn., 110, pp. 841-855.
Shahna, K. U., (2023), Novel chaos based cryptosystem using four-dimensional hyper chaotic map with efficient permutation and substitution techniques, Chaos, Solitons & Fractals, 170, pp. 113383.
Zhou, X., Sun, K., Wang, H., et al., (2024), Coexisting hyperchaos and multistability in a discrete memristor-coupled bi-neuron model, Nonlinear Dyn., 112, pp. 9547-9561.
Lorenz, E. N., (1963), Deterministic nonperiodic flow, J. Atmos. Sci., 20(2), pp. 130-141.
Rössler, O. E., (1976), An equation for continuous chaos, Phys. Lett. A, 57(5), pp. 397-398.

Rössler, O. E., (1979), Continuous Chaos – Four Prototype Equations, Ann. NY Acad. Sci., 316, pp.376-392.
Sprott, J. C., (2023), Elegant Automation, WORLD SCIENTIFIC, 348 p.
Leonov, G. A., Kuznetsov, N. V. and Mokaev, T. N., (2015), Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion, Eur. Phys. J. Special Topics, 224, pp. 1421-1458.
Leonov, G. A., Kuznetsov, N. V. Vagaitsev, V. I., (2011), Localization of hidden Chia’s attractors, Phys. Lett. A, 375(23), pp. 2230-2233.
Al-Azzawi, S. F., Hasan, A. M., (2024), A New 4D Hidden Hyperchaotic System with Higher Largest Lyapunov exponent and its Synchronization, Int. J. Math. Comput. Sci., 2, pp. 63-74.
Zhang, S., Zeng, Y., (2019), A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees, Chaos, Solitons & Fractals, 120, pp. 25-40.
Zhu, X., Du, W. S., (2019), A new family of chaotic systems with different closed curve equilibrium, Mathematics, 7(1), pp. 94.
Bayani, A., Rajagopal, K., Khalaf, A. J. M., Jafari, S., Leutcho, G. D. and Kengne, J., (2019), Dynamical analysis of a new multistable chaotic system with hidden attractor: Antimonotonicity, coexisting multiple attractors, and offset boosting, Phys. Lett. A, 383, pp. 1450-1456.
Nazarimehr, F., Rajagopal, K., Kengne, J., Jafari, S. and Pham, V. T., (2018), A new four-dimensional system containing chaotic or hyper-chaotic attractors with no equilibrium, a line of equilibria and unstable equilibria, Chaos, Solitons & Fractals, 111, pp. 108-118.
Deng, Q., Wang, C., and Yang, L., (2020), Four-wing hidden attractors with one stable equilibrium point, Int. J. Bifurc. Chaos, 30, pp. 2050086.
Jung, W., Elliot, S. J., and Cheer, J., (2019), Local active control of road noise inside a vehicle, Mech. Syst. Signal Process., 121, pp. 144-157.
Bhat, M. A. and Shikha, (2019), Complete synchronisation of non-identical fractional order hyper-chaotic systems using active control, International Journal of Automation and Control, 13, pp. 140-157.
Zhang, H., Zhang, W., Zhao, Y. and Ji, M., (2020), Adaptive state observers for incrementally quadratic nonlinear systems with application to chaos synchronization, Circuits Syst. Signal Process., 39, pp. 1290-1306.
Tohidi, S., Yildiz, Y. and Kolmanovsky, I., (2020), Adaptive control allocation for constrained systems, Automatica, 121, pp. 1-11.
Vaidyanathan, S., Volos, C. K. and Pham, V. T., (2014), Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation, Arch. Control Sci., 24, pp. 409-446.
Vaidyanathan, S. and Volos, C. K., (2015), Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system, Arch. Control Sci., 25, pp. 333-353.
Chu, J. and Hu, W. W., (2016), Control chaos for permanent magnet synchronous motor base on adaptive backstepping of error compensation, Int. J. Autom. Comput., 9, pp. 163-174.
Rajagopal, K., Guessas, L., Vaidyanathan, S., Karthikeyan, A. and Srinivasan, A., (2017), Dynamical analysis and FPGA implementation of a novel hyperchaotic system and its synchronization using adaptive sliding mode control and genetically optimized PID control, Math. Probl. Eng., 2017, pp. 1-14.
Rajagopal, K., Laarem, G., Karthikeyan, A. and Srinivasan, A., (2017), FPGA implementation of adaptive sliding mode control and genetically optimized PID control for fractional-order induction motor system with uncertain load, Adv. Differ. Equ., 2017, pp. 1-20.
Yousefpour, A., Hosseinloo, A. H., Yazdi, M. R. H. and Bahrami, A., (2020), Disturbance observer-based terminal sliding mode control for effective performance of a nonlinear vibration energy harvester, J. Intell. Mater. Syst. Struct., 31, pp. 1495-1510.
Singh, J. P., Roy, B., (2017), Coexistence of asymmetric hidden chaotic attractors in a new simple 4-D chaotic system with curve of equilibria, Optik, 145, pp. 209-217.
Singh, J. P. and Roy, B. K., (2018), Multistability and hidden chaotic attractors in a new simple 4-D chaotic system with chaotic 2-torus behavior, Int. J. Dyn. Contr., 6(2), pp. 529-538.
Zhang, S., Zeng, Y., Li, Z., Wang, M., Zhang, X. and Chang, D., (2018), A novel simple no-equilibrium chaotic system with complex hidden dynamics, Int. J. Dyn. Contr., 6(4), pp. 1465-1476.
Signing, V., Kengne, J., (2018), Coexistence of hidden attractors, 2-torus and 3-torus in a new simple 4-D chaotic system with hyperbolic cosine nonlinearity, Int. J. Dyn. Control, 6, pp. 1421-1428.
Sahin, M. E., Cam Taskiran, Z. G., Guler, H., Hamamci, S. E., (2020), Application and modeling of a novel 4D memristive chaotic system for communication systems, Circuits Syst. Signal Process., 39, pp. 3320-3349.
Bouteraa, Y., Mostafaee, J., Kchaou, M., Abbassi, R., Jerbi, H., Mobayen, S., (2022), A New Simple Chaotic System with One Nonlinear Term, Mathematics, 10, pp. 4374.
Folifack Signing, V.R., Kengne, J., Pone, J.R.M., (2019), Antimonotonicity, chaos, quasi-periodicity and coexistence of hidden attractors in a new simple 4-D chaotic system with hyperbolic cosine non-linearity, Chaos Soliton. Fract., 118, pp. 187-198.
Benettin, G., Galgani, L., Giorgilli, A. et al., ( 1980), Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them, Meccanica, 15, pp. 9-20.
Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A., (1985), Determining Lyapunov Exponents from a Time Series, Physica D, 16, pp.285-317.
Sandri, M., (1996), Numerical Calculation of Lyapunov Exponents, The Mathematica Journal, 6, pp. 78-84.
Singh, J. P. and Roy, B. K., (2016), The nature of Lyapunov exponents is (+, +, −, −). Is it a hyperchaotic system?, Chaos, Solitons & Fractals, 92, pp. 73-85.
Frederickson, P., Kaplan, J. L., Yorke, E. D., Yorke, J. A., (1983), The Liapunov Dimension of Strange Attractors, Journal of differential equations, 49, pp. 185-207.
Wen, J., Feng, Y., Tao, X. and Cao, Y., (2021), Dynamical Analysis of a New Chaotic System: Hidden Attractor, Coexisting-Attractors, Offset Boosting, and DSP Realization, IEEE Access, 9, pp. 167920-167927.
Deng, Q., Wang, C., Lin, H., (2024), Memristive Hopfield neural network dynamics with heterogeneous activation functions and its application, Chaos, Solitons & Fractals, 178, pp. 114387.
Sedra, A. S. and Smith, K. C., (1998), Microelectronics Circuits, 4th ed. New York Oxford University Press.
Wang, X. and Wang, M., (2008), A hyperchaos generated from Lorenz system, Phys. A: Stat. Mech. Appl., 387, pp. 3751-3758.

Details

Primary Language

English

Subjects

Ordinary Differential Equations, Difference Equations and Dynamical Systems, Dynamical Systems in Applications

Journal Section

Research Article

Authors

Michael Kopp ^*
0000-0001-7457-3272
Ukraine

Publication Date

November 3, 2025

Submission Date

October 16, 2024

Acceptance Date

February 26, 2025

Published in Issue

Year 2025 Volume: 15 Number: 11

IZ

https://izlik.org/JA24CC75RP

Cite

RIS / Bibtex

APA

Kopp, M. (2025). CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM. TWMS Journal of Applied and Engineering Mathematics, 15(11), 2584-2598. https://izlik.org/JA24CC75RP

AMA

1.Kopp M. CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM. JAEM. 2025;15(11):2584-2598. https://izlik.org/JA24CC75RP

Chicago

Kopp, Michael. 2025. “CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM”. TWMS Journal of Applied and Engineering Mathematics 15 (11): 2584-98. https://izlik.org/JA24CC75RP.

EndNote

Kopp M (November 1, 2025) CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM. TWMS Journal of Applied and Engineering Mathematics 15 11 2584–2598.

IEEE

[1]M. Kopp, “CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM”, JAEM, vol. 15, no. 11, pp. 2584–2598, Nov. 2025, [Online]. Available: https://izlik.org/JA24CC75RP

ISNAD

Kopp, Michael. “CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM”. TWMS Journal of Applied and Engineering Mathematics 15/11 (November 1, 2025): 2584-2598. https://izlik.org/JA24CC75RP.

JAMA

1.Kopp M. CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM. JAEM. 2025;15:2584–2598.

MLA

Kopp, Michael. “CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM”. TWMS Journal of Applied and Engineering Mathematics, vol. 15, no. 11, Nov. 2025, pp. 2584-98, https://izlik.org/JA24CC75RP.

Vancouver

1.Michael Kopp. CHAOS AND COMPLEXITY IN A FOUR-DIMENSIONAL SYSTEM WITH HYPERBOLIC TANGENT NONLINEARITY AND NO EQUILIBRIUM. JAEM [Internet]. 2025 Nov. 1;15(11):2584-98. Available from: https://izlik.org/JA24CC75RP