Hidden Attractors in Chaotic Systems with Nonlinear Functions

Year 2024, Volume: 6 Issue: 2, 144 - 151, 30.06.2024

Hafiz Muhammad Zeeshan , Rider Jaimes-reategui , Juan Hugo García López , Safara Bibi , Guillermo Huerta-cuellar

https://doi.org/10.51537/chaos.1381891

Abstract

In the present work, an interesting mini-review of hidden attractors in dynamical systems with associated nonlinear functions is carried out. Chaotic systems with nonlinear functions often possess hidden attractors due to their inherent complexity. These attractors can arise in various mathematical models, such as the Lorenz system, Rössler system, or Chua's circuit. The identification and comprehension of hidden attractors broaden our understanding of complex systems and provide new directions for future study and technological development. The discovery and characterization of hidden attractors in chaotic systems have profound implications for various scientific disciplines, including physics, biology, and engineering.

Keywords

Hidden attractors, Chaotic systems, Nonlinear functions, Fixed points, Numerical simulations

Supporting Institution

Consejo Estatal de Ciencia y Tecnología del Estado de Jalisco

Thanks

G.H.C. acknowledges the Fondo de Desarrollo Científico de Jalisco para atender Retos Sociales (2022), for the approved project 10304-2022

References

• Abdolmohammadi, H. R., A. J. M. Khalaf, S. Panahi, K. Rajagopal, V.-T. Pham, et al., 2018 A new 4d chaotic system with hidden attractor and its engineering applications: Analog circuit design and field programmable gate array implementation. Pramana 90: 1–7.
• Ahmadi, A., K. Rajagopal, F. E. Alsaadi, V.-T. Pham, F. E. Alsaadi, et al., 2020 A novel 5d chaotic system with extreme multi-stability and a line of equilibrium and its engineering applications: circuit design and fpga implementation. Iranian Journal of Science and Technology, Transactions of Electrical Engineering 44: 59–67.
• Campos, E. et al., 2020 Multistable systems with hidden and selfexcited scroll attractors generated via piecewise linear systems. Complexity 2020.
• Cang, S., Y. Li, R. Zhang, and Z. Wang, 2019 Hidden and selfexcited coexisting attractors in a lorenz-like system with two equilibrium points. Nonlinear Dynamics 95: 381–390.
• Cao, H.-Y. and L. Zhao, 2021 A new chaotic system with different equilibria and attractors. The European Physical Journal Special Topics 230: 1905–1914.
• Chen, H., S. He, A. D. Pano Azucena, A. Yousefpour, H. Jahanshahi, et al., 2020 A multistable chaotic jerk system with coexisting and hidden attractors: Dynamical and complexity analysis, fpga-based realization, and chaos stabilization using a robust controller. Symmetry 12: 569.
• Djorwe, P., J. Yves Effa, and S. G. Nana Engo, 2023 Hidden attractors and metamorphoses of basin boundaries in optomechanics. Nonlinear Dynamics 111: 5905–5917.
• Dubois, P., T. Gomez, L. Planckaert, and L. Perret, 2020 Datadriven predictions of the lorenz system. Physica D: Nonlinear Phenomena 408: 132495.
• Dudkowski, D., S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, et al., 2016 Hidden attractors in dynamical systems. Physics Reports 637: 1–50.
• Dueñas, J., C. Núñez, and R. Obaya, 2023 Bifurcation theory of attractors and minimal sets in d-concave nonautonomous scalar ordinary differential equations. Journal of Differential Equations 361: 138–182.
• Escalante-González, R. and E. Campos, 2022 Multistable systems with nested hidden and self-excited double scroll attractors. The European Physical Journal Special Topics 231: 351–357.
• Escalante-González, R. and E. Campos-Cantón, 2019 Coexistence of hidden attractors and self-excited attractors through breaking heteroclinic-like orbits of switched systems. arXiv preprint arXiv:1908.03789 .
• Gong, L., R. Wu, and N. Zhou, 2020 A new 4d chaotic system with coexisting hidden chaotic attractors. International Journal of Bifurcation and Chaos 30: 2050142.
• Gong, L.-H., H.-X. Luo, R.-Q.Wu, and N.-R. Zhou, 2022 New 4d chaotic system with hidden attractors and self-excited attractors and its application in image encryption based on rng. Physica A: Statistical Mechanics and its Applications 591: 126793.
• Gray, W. S., M. Palmstrøm, and A. Schmeding, 2023 Continuity of formal power series products in nonlinear control theory. Foundations of Computational Mathematics 23: 803–832.
• Hu, X., C. Liu, L. Liu, Y. Yao, and G. Zheng, 2017 Multi-scroll hidden attractors and multi-wing hidden attractors in a 5- dimensional memristive system. Chinese Physics B 26: 110502.
• Huang, L., W. Yao, J. Xiang, and L. WANG, 2022 Extreme multistability of a four-dimensional chaotic system with infinitely many symmetric homogeneous attractors. 44: 390–399.
• Islam, Y., C. Li, Y. Jiang, X. Ma, A. Akgul, et al., 2022 A hidden chaotic attractor with an independent amplitude-frequency controller. Complexity 2022.
• Jafari, S., A. Ahmadi, A. J. M. Khalaf, H. R. Abdolmohammadi, V.-T. Pham, et al., 2018 A new hidden chaotic attractor with extreme multi-stability. AEU-International Journal of Electronics and Communications 89: 131–135.
• Jafari, S., J. Sprott, and S. M. R. H. Golpayegani, 2013 Elementary quadratic chaotic flows with no equilibria. Physics Letters A 377: 699–702.
• Jasim, B. H., K. H. Hassan, and K. M. Omran, 2021 A new 4-d hyperchaotic hidden attractor system: Its dynamics, coexisting attractors, synchronization and microcontroller implementation. International Journal of Electrical & Computer Engineering (2088-8708) 11.
• Khalaf, A. J. M., H. R. Abdolmohammadi, A. Ahmadi, L. Moysis, C. Volos, et al., 2020 Extreme multi-stability analysis of a novel 5d chaotic system with hidden attractors, line equilibrium, permutation entropy and its secure communication scheme. The European Physical Journal Special Topics 229: 1175–1188.
• Kingni, S. T., G. F. Kuiate, V. K. Tamba, V.-T. Pham, and D. V. Hoang, 2019 Self-excited and hidden attractors in an autonomous josephson jerk oscillator: analysis and its application to text encryption. Journal of Computational and Nonlinear Dynamics 14: 071004.
• Kuznetsov, N., 2020 Theory of hidden oscillations and stability of control systems. Journal of Computer and Systems Sciences International 59: 647–668.
• Kuznetsov, N., T. Mokaev, V. Ponomarenko, E. Seleznev, N. Stankevich, et al., 2023 Hidden attractors in chua circuit: mathematical theory meets physical experiments. Nonlinear Dynamics 111: 5859–5887.
• Kuznetsov, N. V., M. Y. Lobachev, M. V. Yuldashev, R. V. Yuldashev, E. V. Kudryashova, et al., 2020 The birth of the global stability theory and the theory of hidden oscillations. In 2020 European Control Conference (ECC), pp. 769–774, IEEE.
• Lai, Q., Z. Wan, and P. D. Kamdem Kuate, 2020 Modelling and circuit realisation of a new no-equilibrium chaotic system with hidden attractor and coexisting attractors. Electronics Letters 56: 1044–1046.
• Lakshmanan, M. and S. Rajaseekar, 2012 Nonlinear dynamics: integrability, chaos and patterns. Springer Science & Business Media.
• Lin, H., C.Wang, and Y. Tan, 2020 Hidden extreme multistability with hyperchaos and transient chaos in a hopfield neural network affected by electromagnetic radiation. Nonlinear Dynamics 99: 2369–2386.
• Molaie, M., S. Jafari, J. C. Sprott, and S. M. R. H. Golpayegani, 2013 Simple chaotic flows with one stable equilibrium. International Journal of Bifurcation and Chaos 23: 1350188.
• Munmuangsaen, B. and B. Srisuchinwong, 2018 A hidden chaotic attractor in the classical lorenz system. Chaos, Solitons & Fractals 107: 61–66.
• Nag Chowdhury, S. and D. Ghosh, 2020 Hidden attractors: A new chaotic system without equilibria. The European Physical Journal Special Topics 229: 1299–1308.
• Njitacke, Z., T. Fozin, L. K. Kengne, G. Leutcho, E. M. Kengne, et al., 2020 Multistability and its annihilation in the chua’s oscillator with piecewise-linear nonlinearity. Chaos Theory and Applications 2: 77–89.
• Pham, V.-T., C. Volos, S. Jafari, and T. Kapitaniak, 2017 Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dynamics 87: 2001–2010.
• Pulido-Luna, J. R., J. A. López-Rentería, N. R. Cazarez-Castro, and E. Campos, 2021 A two-directional grid multiscroll hidden attractor based on piecewise linear system and its application in pseudo-random bit generator. Integration 81: 34–42.
• Rybin, V., A. Tutueva, T. Karimov, G. Kolev, D. Butusov, et al., 2021 Optimizing the synchronization parameters in adaptive models of rössler system. In 2021 10th Mediterranean Conference on Embedded Computing (MECO), pp. 1–4, IEEE.
• Tanaka, K., T. Ikeda, and H. O. Wang, 1998 A unified approach to controlling chaos via an lmi-based fuzzy control system design. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 45: 1021–1040.
• Wang, N., G. Zhang, N. V. Kuznetsov, and H. Bao, 2021 Hidden attractors and multistability in a modified chua’s circuit. Communications in Nonlinear Science and Numerical Simulation 92: 105494.
• Wang, X. and G. Chen, 2012 A chaotic system with only one stable equilibrium. Communications in Nonlinear Science and Numerical Simulation 17: 1264–1272.
• Wei, Z., 2011 Dynamical behaviors of a chaotic system with no equilibria. Physics Letters A 376: 102–108.
• Wu, X., H. Wang, and S. He, 2021 Localization of hidden attractors in chua’s system with absolute nonlinearity and its fpga implementation. Frontiers in Physics 9: 788329.
• Yang, L. and Q. Lai, 2023 Construction and implementation of discrete memristive hyperchaotic map with hidden attractors and self-excited attractors. Integration p. 102091.
• Ye, X. and X. Wang, 2023 Hidden oscillation and chaotic sea in a novel 3d chaotic system with exponential function. Nonlinear Dynamics pp. 1–10.
• Zaqueros-Martinez, J., G. Rodriguez-Gomez, E. Tlelo-Cuautle, and F. Orihuela-Espina, 2023 Fuzzy synchronization of chaotic systems with hidden attractors. Entropy 25: 495.
• Zelinka, I., 2016 Evolutionary identification of hidden chaotic attractors. Engineering Applications of Artificial Intelligence 50: 159–167.