Chaotic oscillations in a new 2-D discrete dynamical system with hidden parameter
DOI:
https://doi.org/10.5269/bspm.77129Resumo
The main purpose of this work is to present a new 2-D chaotic discrete dynamical system. By studying its basic properties, such as determining its fixed points and their stability types based on bifurcation parameter values, and using Lyapunov exponents, we identify chaotic oscillations for certain values of the parameter a, regardless of the value of the second parameter b. This implies that b has no effect on the system’s dynamics, making it a distinctive feature, henceforth referred to as the hidden parameter. The system, denoted by B_{a,b}, exhibits several useful characteristics, including its quadratic form, the differentiability of its corresponding function, and the absence of the parameter b in its eigenvalues, allowing for arbitrary selection of b and simplified calculations. Simulation results validate the chaotic behavior.
Referências
2. Boukhalfa El-hafsi and Zeraoulia Elhadj, Existence of super chaotic attractors in a general piecewise smooth map of the plane, Interdiscip. Descr. Complex Syst. 12 (2014), no. 1, 92–98.
3. Chai, W. and Chua, L. O., A simple way to synchronize chaotic systems with applications to secure communication systems, Int. J. Bifurcation Chaos 3 (1993), 1619–1625.
4. Everett, S., On the use of dynamical systems in cryptography, 2024.
5. Garcia-Bosque, M., Perez-Resa, A., Sanchez-Azqueta, C., Aldea, C. and Celma, S., Chaos-based bitwise dynamical pseudorandom number generator on FPGA, IEEE Trans. Instrum. Meas. 68 (2018), no. 1, 291–293.
6. Henon, M., A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), 69–77.
7. Kocarev, L., Chaos-based cryptography: a brief overview, IEEE Circuits Syst. Mag. 1 (2001), no. 3, 6–21.
8. Leuciuc, A., Information transmission using chaotic discrete-time filter, IEEE Trans. Circuits Syst. I 47 (2000), no. 1, 82–88.
9. Lorenz, E. N., Deterministic nonperiodic flow, J. Atmospheric Sci. 20 (1963), 130–141.
10. Liu, S., Akashi, N., Huang, Q., Kuniyoshi, Y. and Nakajima, K., Exploiting chaotic dynamics in deep neural networks, 2024.
11. Lozi, R., Un attracteur etrange du type attracteur de Henon, J. Phys. 39 (1978), 9–10.
12. May, R. M., Simple mathematical models with very complicated dynamics, Nature 261 (1976), no. 5560, 459–467.
13. Maaita, J. and Prousalis, D., A comparison between four chaotic indicators in systems with hidden attractors, J. Comput. Nonlinear Dynam. 20 (2024), no. 1, 011008, https://doi.org/10.1115/1.4067010.
14. Mogharabin, N. and Ghadami, A., Bifurcation analysis in dynamical systems through integration of machine learning and dynamical systems theory, J. Comput. Nonlinear Dynam. 20 (2024), no. 2, 021006, https://doi.org/10.1115/1.4067297.
15. Williamson, A., Chaos, CRC Press, 2022.
16. Yoshida, H., Kurata, S., Li, Y. and Nara, S., Chaotic neural network applied to two-dimensional motion control, Cogn. Neurodyn. 4 (2010), no. 1, 69–80, https://doi.org/10.1007/s11571-009-9101-5.
17. Zang, X., Iqbal, S., Zhu, Y., Liu, X. and Zhao, J., Applications of chaotic dynamics in robotics, Int. J. Adv. Robot. Syst. 13 (2016), no. 2, 60.
18. Zeraoulia Elhadj, A new chaotic attractor from 2-D discrete mapping via border-collision period doubling scenario, Discrete Dyn. Nat. Soc. 2005 (2005), 235–238.
Downloads
Publicado
Edição
Seção
Licença
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
