Hopf Bifurcation Analysis of the Halvorsen System

Kardo Baiz Othman; Adnan Ali Jalal

doi:10.21271/ZJPAS.37.6.4

Authors

Kardo Baiz Othman Department of Physics, College of Educations-Shaqlawa, Salahaddin University-Erbil, Erbil, Iraq.
Adnan Ali Jalal Department of Mathematics, College of Education, Salahaddin University-Erbil, Erbil, Iraq.

DOI:

https://doi.org/10.21271/ZJPAS.37.6.4

Keywords:

Halvorsen system, Hopf Bifurcation, Transcritical bifurcation, Periodic orbits, Normal form theory

Abstract

This paper investigates local bifurcations in the Halvorsen system, focusing specifically on transcritical and Hopf bifurcations. The behavior of equilibrium points during bifurcations is studied using Sotomayor's theorem for transcritical bifurcation and normal form theory, which is based on Hassard's formulas, for Hopf bifurcation. When the bifurcation parameter exceeds a critical value, a Hopf bifurcation emerges. By applying normal form theory, we establish the conditions under which a Hopf bifurcation occurs. Furthermore, we discuss the direction of the Hopf bifurcation and the stability of the resulting periodic orbits. Finally, numerical simulations are provided to support the theoretical findings.

References

Berir, M. 2024. Analysis of the Effect of White Noise on the Halvorsen System of Variable-Order Fractional Derivatives Using a Novel Numerical Method. International Journal of Advances in Soft Computing and its Applications, 16, 294-306.

Chen, H.-K., Lee, C.-I. & Fractals. 2004. Anti-control of chaos in rigid body motion. Chaos, Solitons, 21, 957-965.

Colak, B., Karaaslan, M., Alkurt, F. O., Bakir, M., Akdogan, V., Oral, M. & Koksal, A. S. 2024. Halvorsen chaotic system based microwave absorber modelling for fighter jet stealth technologies. Optik, 317, 172075.

Feng, L., Liu, Y., Shi, B. & Liu, J. 2025. Toward a physics-guided machine learning approach for predicting chaotic systems dynamics. Frontiers in Big Data, 7, 1506443.

Gotthans, T. & Petrzela, J. 2011. Novel quantification for chaotic dynamical systems with large state attractors. Recent Researches in Mathematical Methods in Electrical Engineering

Computer Science.

Hammouch, Z., Yavuz, M. & Özdemir, N. 2021. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1, 11-23.

Hassard, B. & Wan, Y. H. 1978. Bifurcation formulae derived from center manifold theory. Journal of Mathematical Analysis Applications, 63, 297-312.

Hassard, B. D., Kazarinoff, N. D. & Wan, Y.-H. 1981. Theory and applications of Hopf bifurcation, CUP Archive.

Herteux, J. & Räth, C. 2020. Breaking symmetries of the reservoir equations in echo state networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30.

Husien, A. M. & Amen, A. I. 2024. Hopf and Zero-Hopf Bifurcation Analysis for a Chaotic System. 34, 2450104.

Kuznetsov, Y. A., Kuznetsov, I. A. & Kuznetsov, Y. 1998. Elements of applied bifurcation theory, Springer.

Lorenz, E. N. 1963. The mechanics of vacillation. Journal of Atmospheric Sciences, 20, 448-465.

Lü, J. & Chen, G. 2002. A new chaotic attractor coined. International Journal of Bifurcation chaos, 12, 659-661.

Lynch, S. 2004. Dynamical systems with applications using MATLAB, Springer.

Ma, H., Haluszczynski, A., Prosperino, D. & Räth, C. 2022. Identifying causality drivers and deriving governing equations of nonlinear complex systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32.

Miao, H., Zhu, W., Dan, Y. & Yu, N. 2024. Chaotic time series prediction based on multi-scale attention in a multi-agent environment. Chaos, Solitons Fractals, 183, 114875.

Mirkhan, J. M. & Amen, A. I. 2022. Bifurcation analysis for Shil'nikov Chaos Electro-dissolution of Copper. Zanco Journal of Pure and Applied Sciences, 34, 83-91.

Perko, L. 2013. Differential equations and dynamical systems, Springer Science & Business Media.

Petržela, J., Hruboš, Z. & Gotthans, T. 2011. Modeling Deterministic Chaos Using Electronic Circuits. Radioengineering, 20.

Rössler, O. E. 1976. An equation for continuous chaos. Physics Letters A, 57, 397-398.

Salih, H. R. & Mohhamad, B. 2024. Stability and Hopf bifurcation in a modified Sprott C system. Tatra Mt. Math. Publ, 88, 59-72.

Sprott, J. 2003. Chaos and Time-Series Analysis. Oxford University Press.

Sprott, J. C. 2010. Elegant chaos: algebraically simple chaotic flows, World Scientific.

Sprott, J. C. 2023. Elegant Automation: Robotic Analysis of Chaotic Systems, World Scientific.

Strogatz, S. H. 2024. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Chapman and Hall/CRC.

Thomas, R. 1999. Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis," labyrinth chaos". International Journal of Bifurcation and Chaos, 9, 1889-1905.

Tutueva, A., Andreev, V., Karimov, T., Kopets, E. & Khalyasmaa, A. Fixed-Point Implementation of Extrapolation ODE Solvers. 2019 Ural Symposium on Biomedical Engineering, Radioelectronics and Information Technology (USBEREIT), 2019. IEEE, 310-312.

Vaidyanathan, S. & Azar, A. T. 2016. Adaptive control and synchronization of Halvorsen circulant chaotic systems. Advances in chaos theory and intelligent control. Springer.

Vaidyanathan, S. & Pakiriswamy, S. 2014. Adaptive Controller Design for the Generalized Projective Synchronization of Circulant Chaotic Systems with Unknown Parameters. International Journal of Control Theory and Applications, 7, 55-74.

Wei, Z. & Yang, Q. 2010. Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci. Applied Mathematics Computation, 217, 422-429.

Wu, S.-X. & Meng, X.-Y. 2025. Hopf bifurcation analysis of a multiple delays stage-structure predator-prey model with refuge and cooperation. Electronic Research Archive, 33.

Yousfi, H., Islam, Y., He, S., Gasri, A. & Hassan, M. M. 2024. Advanced medical image encryption techniques using the fractional-order Halvorsen circulant systems: dynamics, control, synchronization and security applications. Physica Scripta, 99, 055208.

Zhou, W., Xu, Y., Lu, H. & Pan, L. 2008. On dynamics analysis of a new chaotic attractor. J Physics Letters A, 372, 5773-5777.

Zhu, C., Liu, Y. & Guo, Y. 2010. Theoretic and numerical study of a new chaotic system. J Intelligent Information Management, 2, 104-109.