IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v175y2023ip2s0960077923009074.html
   My bibliography  Save this article

The global dynamics of a new fractional-order chaotic system

Author

Listed:
  • Liu, Ping
  • Zhang, Yulan
  • Mohammed, Khidhair Jasim
  • Lopes, António M.
  • Saberi-Nik, Hassan

Abstract

This paper investigates the global dynamics of a new 3-dimensional fractional-order (FO) system that presents just cross-product nonlinearities. Firstly, the FO forced Lorenz-84 system is introduced and the stability of its equilibrium points, as well as the chaos control for their stabilization, are addressed. Secondly, dynamical behavior is further analyzed and bifurcation diagrams, phase portraits, and largest Lyapunov exponent (LE) are discussed. Then, the global Mittag-Leffler attractive sets (MLASs) and Mittag-Leffler positive invariant sets (MLPISs) of the FO forced Lorenz-84 system are presented. Finally, the Hamilton energy function (HEF) of the Lorenz-84 system is calculated by using the Helmholtz theorem. The calculation of the Hamilton energy has an essential role on the estimation of chaos in dynamical systems, the guidance of orbits, and stability. In fact, any control action on the dynamical system completely changes the HEF. Numerical simulations are presented for illustrating the theoretical findings.

Suggested Citation

  • Liu, Ping & Zhang, Yulan & Mohammed, Khidhair Jasim & Lopes, António M. & Saberi-Nik, Hassan, 2023. "The global dynamics of a new fractional-order chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 175(P2).
    • Handle: RePEc:eee:chsofr:v:175:y:2023:i:p2:s0960077923009074
    DOI: 10.1016/j.chaos.2023.114006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077923009074
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2023.114006?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Gao, Wei & Yan, Li & Saeedi, Mohammadhossein & Saberi Nik, Hassan, 2018. "Ultimate bound estimation set and chaos synchronization for a financial risk system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 154(C), pages 19-33.
    2. Mbakob Yonkeu, R. & David, Afungchui, 2022. "Coherence and stochastic resonance in the fractional-birhythmic self-sustained system subjected to fractional time-delay feedback and Lévy noise," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    3. Peng, Qiu & Jian, Jigui, 2021. "Estimating the ultimate bounds and synchronization of fractional-order plasma chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    4. Jian, Jigui & Wu, Kai & Wang, Baoxian, 2020. "Global Mittag-Leffler boundedness and synchronization for fractional-order chaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    5. Zhou, Ping & Hu, Xikui & Zhu, Zhigang & Ma, Jun, 2021. "What is the most suitable Lyapunov function?," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    6. Wang, Haijun & Dong, Guili, 2019. "New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 272-286.
    7. Chien, Fengsheng & Inc, Mustafa & Yosefzade, Hamidreza & Saberi Nik, Hassan, 2021. "Predicting the chaos and solution bounds in a complex dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    8. Mahmoud, Gamal M. & Arafa, Ayman A. & Abed-Elhameed, Tarek M. & Mahmoud, Emad E., 2017. "Chaos control of integer and fractional orders of chaotic Burke–Shaw system using time delayed feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 680-692.
    9. Hu Wang & Yongguang Yu & Guoguang Wen, 2014. "Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-15, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ren, Lei & Lin, Ming-Hung & Abdulwahab, Abdulkareem & Ma, Jun & Saberi-Nik, Hassan, 2023. "Global dynamical analysis of the integer and fractional 4D hyperchaotic Rabinovich system," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    2. Lin, Jinchai & Fan, Ruguo & Tan, Xianchun & Zhu, Kaiwei, 2021. "Dynamic decision and coordination in a low-carbon supply chain considering the retailer's social preference," Socio-Economic Planning Sciences, Elsevier, vol. 77(C).
    3. Runzi Luo & Jiaojiao Fu & Haipeng Su, 2019. "The Exponential Stabilization of a Class of n-D Chaotic Systems via the Exact Solution Method," Complexity, Hindawi, vol. 2019, pages 1-7, May.
    4. Chien, Fengsheng & Inc, Mustafa & Yosefzade, Hamidreza & Saberi Nik, Hassan, 2021. "Predicting the chaos and solution bounds in a complex dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    5. Xie, Ying & Zhou, Ping & Yao, Zhao & Ma, Jun, 2022. "Response mechanism in a functional neuron under multiple stimuli," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 607(C).
    6. Tavakolpour-Saleh, A.R., 2021. "A novel theorem on motion stability," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    7. Njitacke, Zeric Tabekoueng & Ramadoss, Janarthanan & Takembo, Clovis Ntahkie & Rajagopal, Karthikeyan & Awrejcewicz, Jan, 2023. "An enhanced FitzHugh–Nagumo neuron circuit, microcontroller-based hardware implementation: Light illumination and magnetic field effects on information patterns," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    8. Zhang, Hai & Cheng, Yuhong & Zhang, Hongmei & Zhang, Weiwei & Cao, Jinde, 2022. "Hybrid control design for Mittag-Leffler projective synchronization on FOQVNNs with multiple mixed delays and impulsive effects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 197(C), pages 341-357.
    9. Ma, Xiaowen & Xu, Ying, 2022. "Taming the hybrid synapse under energy balance between neurons," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    10. Zhou, Ping & Ma, Jun & Xu, Ying, 2023. "Phase synchronization between neurons under nonlinear coupling via hybrid synapse," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    11. Lei, Yong & Xu, Xin-Jian & Wang, Xiaofan & Zou, Yong & Kurths, Jürgen, 2023. "A new criterion for optimizing synchrony of coupled oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    12. Shi, Jianping & He, Ke & Fang, Hui, 2022. "Chaos, Hopf bifurcation and control of a fractional-order delay financial system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 348-364.
    13. Qiao, Zijian & He, Yuanbiao & Liao, Changrong & Zhu, Ronghua, 2023. "Noise-boosted weak signal detection in fractional nonlinear systems enhanced by increasing potential-well width and its application to mechanical fault diagnosis," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    14. Njitacke, Zeric Tabekoueng & Ramakrishnan, Balamurali & Rajagopal, Karthikeyan & Fonzin Fozin, Théophile & Awrejcewicz, Jan, 2022. "Extremely rich dynamics of coupled heterogeneous neurons through a Josephson junction synapse," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    15. Uğur Erkin Kocamaz & Alper Göksu & Harun Taşkın & Yılmaz Uyaroğlu, 2021. "Control of chaotic two-predator one-prey model with single state control signals," Journal of Intelligent Manufacturing, Springer, vol. 32(6), pages 1563-1572, August.
    16. Peng, Qiu & Jian, Jigui, 2021. "Estimating the ultimate bounds and synchronization of fractional-order plasma chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    17. Li, Wang & Zhao, Lingzhi & Shi, Hongjun & Zhao, Donghua & Sun, Yongzheng, 2021. "Realizing generalized outer synchronization of complex dynamical networks with stochastically adaptive coupling," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 379-390.
    18. Muhammad Moin & Waqas Ahmed & Muhammad Rehan & Muhammad Iqbal & Nasim Ullah & Kamran Zeb & Waqar Uddin, 2022. "A Novel Distributed Consensus-Based Approach to Solve the Economic Dispatch Problem Incorporating the Valve-Point Effect and Solar Energy Sources," Energies, MDPI, vol. 16(1), pages 1-23, December.
    19. Ma, Jun, 2024. "Energy function for some maps and nonlinear oscillators," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    20. Yuemin Zheng & Jin Tao & Qinglin Sun & Hao Sun & Zengqiang Chen & Mingwei Sun, 2023. "Adaptive Active Disturbance Rejection Load Frequency Control for Power System with Renewable Energies Using the Lyapunov Reward-Based Twin Delayed Deep Deterministic Policy Gradient Algorithm," Sustainability, MDPI, vol. 15(19), pages 1-25, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:175:y:2023:i:p2:s0960077923009074. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.