Yıl 2024,
Cilt: 6 Sayı: 2, 144 - 151, 30.06.2024
Hafiz Muhammad Zeeshan
Rider Jaimes-reategui
Juan Hugo García López
Safara Bibi
Guillermo Huerta-cuellar
Kaynakça
- 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.
Hidden Attractors in Chaotic Systems with Nonlinear Functions
Yıl 2024,
Cilt: 6 Sayı: 2, 144 - 151, 30.06.2024
Hafiz Muhammad Zeeshan
Rider Jaimes-reategui
Juan Hugo García López
Safara Bibi
Guillermo Huerta-cuellar
Öz
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.
Destekleyen Kurum
Consejo Estatal de Ciencia y Tecnología del Estado de Jalisco
Teşekkür
G.H.C. acknowledges the Fondo de Desarrollo Científico de Jalisco para atender Retos Sociales (2022), for the approved project 10304-2022
Kaynakça
- 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.