Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system

Jie Liu; Bo Sang; Lihua Fan; Chun Wang; Xueqing Liu; Ning Wang; Irfan Ahmad; Jie Liu; Bo Sang; Lihua Fan; Chun Wang; Xueqing Liu; Ning Wang; Irfan Ahmad

doi:10.3934/math.2025225

AIMS Mathematics

2025, Volume 10, Issue 3: 4915-4937. doi: 10.3934/math.2025225

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Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system

1.
School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, Shandong, China
2.
School of Microelectronics and Control Engineering, Changzhou University, Changzhou 213159, Jiangsu, China
3.
Faculty of Science and Engineering, Maynooth International Engineering College, Maynooth University, Maynooth, Co. Kildare W23, Ireland

Received: 23 December 2024 Revised: 02 February 2025 Accepted: 17 February 2025 Published: 06 March 2025
MSC : 37D45, 37G35

Chameleon systems are dynamical systems that exhibit either self-excited or hidden oscillations depending on the parameter values. This paper presents a comprehensive investigation of a quadratic chameleon system, including an analysis of its symmetry, dissipation, local stability, Hopf bifurcation, and various chaotic dynamics as the control parameters $ (\mu, a, c) $ vary. Here, $ \mu $ serves as the dissipation parameter in the $ y $ direction. Bifurcation analysis for four scenarios with $ \mu = 0 $ was performed, revealing the emergence of various dynamical phenomena under different parameter settings. Offset boosting means introducing a constant into one of the state variables of the system for boosting the variable to a different level. Additionally, hidden chaotic bistability with offset boosting was exhibited by varying $ \mu $. The parameter $ \mu $ serves as both the Hopf bifurcation parameter and the offset boosting parameter, while the other parameters $ (a, c) $ also play critical roles as control parameters, resulting in period-doubling routes to self-excited or hidden chaotic attractors. These findings enrich our understanding of nonlinear dynamics in quadratic chameleon systems.
- Hopf bifurcation,
- antimonotonicity,
- bifurcation diagram,
- hidden chaotic attractor,
- offset boosting
Citation: Jie Liu, Bo Sang, Lihua Fan, Chun Wang, Xueqing Liu, Ning Wang, Irfan Ahmad. Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system[J]. AIMS Mathematics, 2025, 10(3): 4915-4937. doi: 10.3934/math.2025225

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Abstract

Chameleon systems are dynamical systems that exhibit either self-excited or hidden oscillations depending on the parameter values. This paper presents a comprehensive investigation of a quadratic chameleon system, including an analysis of its symmetry, dissipation, local stability, Hopf bifurcation, and various chaotic dynamics as the control parameters $ (\mu, a, c) $ vary. Here, $ \mu $ serves as the dissipation parameter in the $ y $ direction. Bifurcation analysis for four scenarios with $ \mu = 0 $ was performed, revealing the emergence of various dynamical phenomena under different parameter settings. Offset boosting means introducing a constant into one of the state variables of the system for boosting the variable to a different level. Additionally, hidden chaotic bistability with offset boosting was exhibited by varying $ \mu $. The parameter $ \mu $ serves as both the Hopf bifurcation parameter and the offset boosting parameter, while the other parameters $ (a, c) $ also play critical roles as control parameters, resulting in period-doubling routes to self-excited or hidden chaotic attractors. These findings enrich our understanding of nonlinear dynamics in quadratic chameleon systems.

References

[1]	R. A. Meyers, Encyclopedia of physical science and technology, San Diego: Academic Press, 1992.
[2]	C. Li, W. Hu, J. C. Sprott, X. Wang, Multistability in symmetric chaotic systems, Eur. Phys. J. Spec. Top., 224 (2015), 1493–1506. https://doi.org/10.1140/epjst/e2015-02475-x doi: 10.1140/epjst/e2015-02475-x
[3]	T. A. Alexeeva, N. V. Kuznetsov, T. N. Mokaev, Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents, Chaos Soliton. Fract., 152 (2021), 111365. https://doi.org/10.1016/j.chaos.2021.111365 doi: 10.1016/j.chaos.2021.111365
[4]	S. Vaidyanathan, A. S. T. Kammogne, E. Tlelo-Cuautle, C. N. Talonang, B. Abd-El-Atty, A. A. Abd El-Latif, et al., A novel 3-D jerk system, its bifurcation analysis, electronic circuit design and a cryptographic application, Electronics, 12 (2023), 2818. https://doi.org/10.3390/electronics12132818 doi: 10.3390/electronics12132818
[5]	C. Nwachioma, J. H. P'erez-Cruz, Analysis of a new chaotic system, electronic realization and use in navigation of differential drive mobile robot, Chaos Soliton. Fract., 144 (2021), 110684. https://doi.org/10.1016/j.chaos.2021.110684 doi: 10.1016/j.chaos.2021.110684
[6]	N. V. Kuznetsov, G. A. Leonov, V. I. Vagaitsev, Analytical-numerical method for attractor localization of generalized Chua's system, IFAC Proc. Vol., 43 (2010), 29–33. https://doi.org/10.3182/20100826-3-TR-4016.00009 doi: 10.3182/20100826-3-TR-4016.00009
[7]	G. A. Leonov, N. V. Kuznetsov, V. I. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A, 375 (2011), 2230–2233. https://doi.org/10.1016/j.physleta.2011.04.037 doi: 10.1016/j.physleta.2011.04.037
[8]	G. A. Leonov, N. V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurcat. Chaos, 23 (2013), 1330002. https://dx.doi.org/10.1142/S0218127413300024 doi: 10.1142/S0218127413300024
[9]	N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov, L. O. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurcat. Chaos, 27 (2017), 1730038. https://doi.org/10.1142/S0218127417300385 doi: 10.1142/S0218127417300385
[10]	N. Kuznetsov, T. Mokaev, V. Ponomarenko, E. Seleznev, N. Stankevich, L. Chua, Hidden attractors in Chua circuit: mathematical theory meets physical experiments, Nonlinear Dyn., 111 (2023), 5859–5887. https://doi.org/10.1007/s11071-022-08078-y doi: 10.1007/s11071-022-08078-y
[11]	Q. Wu, Q. Hong, X. Liu, X. Wang, Z. Zeng, A novel amplitude control method for constructing nested hidden multi-butterfly and multiscroll chaotic attractors, Chaos Soliton. Fract., 134 (2020), 109727. https://doi.org/10.1016/j.chaos.2020.109727 doi: 10.1016/j.chaos.2020.109727
[12]	H. Tian, Z. Wang, P. Zhang, M. Chen, Y. Wang, Dynamic analysis and robust control of a chaotic system with hidden attractor, Complexity, 2021 (2021), 8865522. https://doi.org/10.1155/2021/8865522 doi: 10.1155/2021/8865522
[13]	F. Bao, S. Yu, How to generate chaos from switching system: a saddle focus of index 1 and heteroclinic loop-based approach, Math. Probl. Eng., 2011 (2011), 756462. https://doi.org/10.1155/2011/756462 doi: 10.1155/2011/756462
[14]	T. Zhou, G. Chen, Classification of chaos in 3-D autonomous quadratic systems-Ⅰ: basic framework and methods, Int. J. Bifurcat. Chaos, 16 (2006), 2459–2479. https://doi.org/10.1142/S0218127406016203 doi: 10.1142/S0218127406016203
[15]	Z. Wei, J. C. Sprott, H. Chen, Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium, Phys. Lett. A, 379 (2015), 2184–2187. https://doi.org/10.1016/j.physleta.2015.06.040 doi: 10.1016/j.physleta.2015.06.040
[16]	K. Rajagopal, V. T. Pham, F. R. Tahir, A. Akgul, H. R. Abdolmohammadi, S. Jafari, A chaotic jerk system with non-hyperbolic equilibrium: dynamics, effect of time delay and circuit realisation, Pramana, 90 (2018), 52. https://doi.org/10.1007/s12043-018-1545-x doi: 10.1007/s12043-018-1545-x
[17]	X. Cai, L. Liu, Y. Wang, C. Liu, A 3D chaotic system with piece-wise lines shape non-hyperbolic equilibria and its predefined-time control, Chaos Soliton. Fract., 146 (2021), 110904. https://doi.org/10.1016/j.chaos.2021.110904 doi: 10.1016/j.chaos.2021.110904
[18]	C. Li, W. Hai, Constructing multiwing attractors from a robust chaotic system with non-hyperbolic equilibrium points, Automatika, 59 (2018), 184–193. https://doi.org/10.1080/00051144.2018.1516273 doi: 10.1080/00051144.2018.1516273
[19]	C. Li, J. C. Sprott, Chaotic flows with a single nonquadratic term, Phys. Lett. A, 378 (2014), 178–183. https://doi.org/10.1016/j.physleta.2013.11.004 doi: 10.1016/j.physleta.2013.11.004
[20]	S. Jafari, J. C. Sprott, F. Nazarimehr, Recent new examples of hidden attractors, Eur. Phys. J. Spec. Top., 224 (2015), 1469–1476. https://doi.org/10.1140/epjst/e2015-02472-1 doi: 10.1140/epjst/e2015-02472-1
[21]	X. Wang, G. Chen, A chaotic system with only one stable equilibrium, Commun. Nonlinear Sci., 17 (2012), 1264–1272. https://doi.org/10.1016/j.cnsns.2011.07.017 doi: 10.1016/j.cnsns.2011.07.017
[22]	X. Wang, A. Akgul, S. Cicek, V. T. Pham, D. V. Hoang, A chaotic system with two stable equilibrium points: dynamics, circuit realization and communication application, Int. J. Bifurcat. Chaos, 27 (2017), 1750130. https://doi.org/10.1142/S0218127417501309 doi: 10.1142/S0218127417501309
[23]	P. C. Rech, Self-excited and hidden attractors in a multistable jerk system, Chaos Soliton. Fract., 164 (2022), 112614. https://doi.org/10.1016/j.chaos.2022.112614 doi: 10.1016/j.chaos.2022.112614
[24]	V. T. Pham, S. Jafari, C. Volos, T. Kapitaniak, A gallery of chaotic systems with an infinite number of equilibrium points, Chaos Soliton. Fract., 93 (2016), 58–63. https://doi.org/10.1016/j.chaos.2016.10.002 doi: 10.1016/j.chaos.2016.10.002
[25]	M. Molaie, S. Jafari, J. C. Sprott, S. M. Golpayegani, Simple chaotic flows with one stable equilibrium, Int. J. Bifurcat. Chaos, 23 (2013), 1350188. https://doi.org/10.1142/S0218127413501885 doi: 10.1142/S0218127413501885
[26]	S. Jafari, J. C. Sprott, Simple chaotic flows with a line equilibrium, Chaos Soliton. Fract., 57 (2013), 79–84. https://doi.org/10.1016/j.chaos.2013.08.018 doi: 10.1016/j.chaos.2013.08.018
[27]	S. Kumarasamy, M. Banerjee, V. Varshney, M. D. Shrimali, N. V. Kuznetsov, A. Prasad, Saddle-node bifurcation of periodic orbit route to hidden attractors, Phys. Rev. E, 107 (2023), L052201. https://doi.org/10.1103/PhysRevE.107.L052201 doi: 10.1103/PhysRevE.107.L052201
[28]	R. Balamurali, J. Kengne, R. G. Chengui, K. Rajagopal, Coupled van der Pol and Duffing oscillators: emergence of antimonotonicity and coexisting multiple self-excited and hidden oscillations, Eur. Phys. J. Plus, 137 (2022), 789. https://doi.org/10.1140/epjp/s13360-022-03000-2 doi: 10.1140/epjp/s13360-022-03000-2
[29]	B. Li, B. Sang, M. Liu, X. Hu, X. Zhang, N. Wang, Some jerk systems with hidden chaotic dynamics, Int. J. Bifurcat. Chaos, 33 (2023), 2350069. https://doi.org/10.1142/S0218127423500694 doi: 10.1142/S0218127423500694
[30]	C. Dong, M. Yang, L. Jia, Z. Li, Dynamics investigation and chaos-based application of a novel no-equilibrium system with coexisting hidden attractors, Phys. A, 633 (2024), 129391. https://doi.org/10.1016/j.physa.2023.129391 doi: 10.1016/j.physa.2023.129391
[31]	J. C. Sprott, Strange attractors with various equilibrium types, Eur. Phys. J. Spec. Top., 224 (2015), 1409–1419. https://doi.org/10.1140/epjst/e2015-02469-8 doi: 10.1140/epjst/e2015-02469-8
[32]	M. A. Jafari, E. Mliki, A. Akgul, V. T. Pham, S. T. Kingni, X. Wang, S. Jafari, Chameleon: the most hidden chaotic flow, Nonlinear Dyn., 88 (2017), 2303–2317. https://doi.org/10.1007/s11071-017-3378-4 doi: 10.1007/s11071-017-3378-4
[33]	K. Rajagopal, A. Karthikeyan, P. Duraisamy, Hyperchaotic chameleon: fractional order FPGA implementation, Complexity, 2017 (2017), 8979408. https://doi.org/10.1155/2017/8979408 doi: 10.1155/2017/8979408
[34]	H. Natiq, M. R. Said, M. R. Ariffin, S. He, L. Rondoni, S. Banerjee, Self-excited and hidden attractors in a novel chaotic system with complicated multistability, Eur. Phys. J. Plus, 133 (2018), 557. https://doi.org/10.1140/epjp/i2018-12360-y doi: 10.1140/epjp/i2018-12360-y
[35]	S. Cang, Y. Li, R. Zhang, Z. Wang, Hidden and self-excited coexisting attractors in a Lorenz-like system with two equilibrium points, Nonlinear Dyn., 95 (2019), 381–390. https://doi.org/10.1007/s11071-018-4570-x doi: 10.1007/s11071-018-4570-x
[36]	Q. Yang, Z. Wei, G. Chen, An unusual 3D autonomous quadratic chaotic system with two stable node-foci, Int. J. Bifurcat. Chaos, 20 (2010), 1061–1083. https://doi.org/10.1142/S0218127410026320 doi: 10.1142/S0218127410026320
[37]	V. F. Signing, G. G. Tegue, M. Kountchou, Z. T. Njitacke, N. Tsafack, J. D. Nkapkop, et al., A cryptosystem based on a chameleon chaotic system and dynamic DNA coding, Chaos Soliton. Fract., 155 (2022), 111777. https://doi.org/10.1016/j.chaos.2021.111777 doi: 10.1016/j.chaos.2021.111777
[38]	W. Fan, D. Xu, Z. Chen, N. Wang, Q. Xu, On two-parameter bifurcation and analog circuit implementation of a chameleon chaotic system, Phys. Scr., 99 (2023), 015218. https://doi.org/10.1088/1402-4896/ad1231 doi: 10.1088/1402-4896/ad1231
[39]	A. Tiwari, R. Nathasarma, B. K. Roy, A new time-reversible 3D chaotic system with coexisting dissipative and conservative behaviors and its active non-linear control, J. Franklin I., 361 (2024), 106637. https://doi.org/10.1016/j.jfranklin.2024.01.038 doi: 10.1016/j.jfranklin.2024.01.038
[40]	R. Zhou, Y. Gu, J. Cui, G. Ren, S. Yu, Nonlinear dynamic analysis of supercritical and subcritical Hopf bifurcations in gas foil bearing-rotor systems, Nonlinear Dyn., 103 (2021), 2241–2256. https://doi.org/10.1007/s11071-021-06234-4 doi: 10.1007/s11071-021-06234-4
[41]	N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov, L. O. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurcat. Chaos, 27 (2017), 1730038. https://doi.org/10.1142/S0218127417300385 doi: 10.1142/S0218127417300385
[42]	H. Zhao, Y. Lin, Y. Dai, Hopf bifurcation and hidden attractor of a modified Chua's equation, Nonlinear Dyn., 90 (2017), 2013–2021. https://doi.org/10.1007/s11071-017-3777-6 doi: 10.1007/s11071-017-3777-6
[43]	M. Liu, B. Sang, N. Wang, I. Ahmad, Chaotic dynamics by some quadratic jerk systems, Axioms, 10 (2021), 227. https://doi.org/10.3390/axioms10030227 doi: 10.3390/axioms10030227
[44]	Q. Yang, D. Zhu, L. Yang, A new 7D hyperchaotic system with five positive Lyapunov exponents coined, Int. J. Bifurcat. Chaos, 28 (2018), 1850057. https://doi.org/10.1142/S0218127418500578 doi: 10.1142/S0218127418500578
[45]	Z. Li, K. Chen, Neuromorphic behaviors in a neuron circuit based on current-controlled Chua Corsage Memristor, Chaos Soliton. Fract., 175 (2023), 114017. https://doi.org/10.1016/j.chaos.2023.114017 doi: 10.1016/j.chaos.2023.114017
[46]	Y. Liu, Y. Zhou, B. Guo, Hopf bifurcation, periodic solutions, and control of a new 4D hyperchaotic system, Mathematics, 11 (2023), 2699. https://doi.org/10.3390/math11122699 doi: 10.3390/math11122699
[47]	C. Li, J. C. Sprott, Variable-boostable chaotic flows, Optik, 127 (2016), 10389–10398. https://doi.org/10.1016/j.ijleo.2016.08.046 doi: 10.1016/j.ijleo.2016.08.046
[48]	C. Li, A. Akgul, L. Bi, Y. Xu, C. Zhang, A chaotic jerk oscillator with interlocked offset boosting, Eur. Phys. J. Plus, 139 (2024), 242. https://doi.org/10.1140/epjp/s13360-024-05040-2 doi: 10.1140/epjp/s13360-024-05040-2
[49]	C. Li, X. Wang, G. Chen, Diagnosing multistability by offset boosting, Nonlinear Dyn., 90 (2017), 1335–1341. https://doi.org/10.1007/s11071-017-3729-1 doi: 10.1007/s11071-017-3729-1
[50]	X. Gao, Hamilton energy of a complex chaotic system and offset boosting, Phys. Scr., 99 (2024), 015244. https://doi.org/10.1088/1402-4896/ad1739 doi: 10.1088/1402-4896/ad1739
[51]	X. Zhang, C. Li, T. Lei, H. Fu, Z. Liu, Offset boosting in a memristive hyperchaotic system, Phys. Scr., 99 (2024), 015247. https://doi.org/10.1088/1402-4896/ad156e doi: 10.1088/1402-4896/ad156e
[52]	H. Dang-Vu, C. Delcarte, Bifurcations et Chaos: une introduction à la dynamique contemporaine avec des programmes en Pascal, Fortran et Mathematica, Paris: Ellipses, 2000.
[53]	C. Grebogi, E. Ott, J. A. Yorke, Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics, Science, 238 (1987), 632–638. https://doi.org/10.1126/science.238.4827.632 doi: 10.1126/science.238.4827.632

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