IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v42y2009i2p1181-1189.html
   My bibliography  Save this article

Dynamic analysis of a fractional-order Lorenz chaotic system

Author

Listed:
  • Yu, Yongguang
  • Li, Han-Xiong
  • Wang, Sha
  • Yu, Junzhi

Abstract

The dynamic behaviors of fractional-order differential systems have received increasing attention in recent decades. But many results about fractional-order chaotic systems are attained only by using analytic and numerical methods. Based on the qualitative theory, the existence and uniqueness of solutions for a class of fractional-order Lorenz chaotic systems are investigated theoretically in this paper. The stability of the corresponding equilibria is also argued similarly to the integer-order counterpart. According to the obtained results, the bifurcation conditions of these two systems are significantly different. Numerical solutions, together with simulations, finally verify the correctness of our analysis.

Suggested Citation

  • Yu, Yongguang & Li, Han-Xiong & Wang, Sha & Yu, Junzhi, 2009. "Dynamic analysis of a fractional-order Lorenz chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1181-1189.
  • Handle: RePEc:eee:chsofr:v:42:y:2009:i:2:p:1181-1189
    DOI: 10.1016/j.chaos.2009.03.016
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077909001283
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2009.03.016?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. Wang, Junwei & Zhang, Yanbin, 2006. "Designing synchronization schemes for chaotic fractional-order unified systems," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1265-1272.
    2. Wang, Junwei & Zhou, Tianshou, 2007. "Chaos synchronization based on contraction principle," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 163-170.
    3. Lu, Jun Guo, 2006. "Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 519-525.
    4. Yan, Jianping & Li, Changpin, 2007. "On chaos synchronization of fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 725-735.
    5. Laskin, Nick, 2000. "Fractional market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 482-492.
    6. Lu, Jun Guo, 2006. "Nonlinear observer design to synchronize fractional-order chaotic systems via a scalar transmitted signal," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 359(C), pages 107-118.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Rongwei Guo & Yaru Zhang & Cuimei Jiang, 2021. "Synchronization of Fractional-Order Chaotic Systems with Model Uncertainty and External Disturbance," Mathematics, MDPI, vol. 9(8), pages 1-12, April.
    2. Shukla, Manoj Kumar & Sharma, B.B., 2017. "Backstepping based stabilization and synchronization of a class of fractional order chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 274-284.
    3. Kamal, F.M. & Elsonbaty, A. & Elsaid, A., 2021. "A novel fractional nonautonomous chaotic circuit model and its application to image encryption," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    4. Agrawal, S.K. & Srivastava, M. & Das, S., 2012. "Synchronization of fractional order chaotic systems using active control method," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 737-752.
    5. Hanshuo Qiu & Xiangzi Zhang & Huaixiao Yue & Jizhao Liu, 2023. "A Novel Eighth-Order Hyperchaotic System and Its Application in Image Encryption," Mathematics, MDPI, vol. 11(19), pages 1-29, September.
    6. Deepika, Deepika & Kaur, Sandeep & Narayan, Shiv, 2018. "Uncertainty and disturbance estimator based robust synchronization for a class of uncertain fractional chaotic system via fractional order sliding mode control," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 196-203.
    7. Yildirim, Gokce & Tanyildizi, Erkan, 2023. "An innovative approach based on optimization for the determination of initial conditions of continuous-time chaotic system as a random number generator," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    8. Ahmad, Shabir & Ullah, Aman & Akgül, Ali, 2021. "Investigating the complex behaviour of multi-scroll chaotic system with Caputo fractal-fractional operator," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    9. Wang, Jieyang & Mou, Jun & Xiong, Li & Zhang, Yingqian & Cao, Yinghong, 2021. "Fractional-order design of a novel non-autonomous laser chaotic system with compound nonlinearity and its circuit realization," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    10. Runlong Peng & Cuimei Jiang & Rongwei Guo, 2021. "Partial Anti-Synchronization of the Fractional-Order Chaotic Systems through Dynamic Feedback Control," Mathematics, MDPI, vol. 9(7), pages 1-13, March.
    11. Peng, Yuexi & Sun, Kehui & Peng, Dong & Ai, Wei, 2019. "Dynamics of a higher dimensional fractional-order chaotic map," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 96-107.
    12. Li, Zengshan & Chen, Diyi & Ma, Mengmeng & Zhang, Xinguang & Wu, Yonghong, 2017. "Feigenbaum's constants in reverse bifurcation of fractional-order Rössler system," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 116-123.
    13. Andrade, Dana I. & Specchia, Stefania & Fuziki, Maria E.K. & Oliveira, Jessica R.P. & Tusset, Angelo M. & Lenzi, Giane G., 2024. "Dynamic analysis and SDRE control applied in a mutating autocatalyst with chaotic behavior," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    14. Leng, Xiangxin & Gu, Shuangquan & Peng, Qiqi & Du, Baoxiang, 2021. "Study on a four-dimensional fractional-order system with dissipative and conservative properties," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    15. Silva-Juárez, Alejandro & Tlelo-Cuautle, Esteban & de la Fraga, Luis Gerardo & Li, Rui, 2021. "Optimization of the Kaplan-Yorke dimension in fractional-order chaotic oscillators by metaheuristics," Applied Mathematics and Computation, Elsevier, vol. 394(C).

    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. Deng, Hongmin & Li, Tao & Wang, Qionghua & Li, Hongbin, 2009. "A fractional-order hyperchaotic system and its synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 962-969.
    2. Sharma, Vivek & Shukla, Manoj & Sharma, B.B., 2018. "Unknown input observer design for a class of fractional order nonlinear systems," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 96-107.
    3. Ge, Zheng-Ming & Yi, Chang-Xian, 2007. "Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 42-61.
    4. Radwan, A.G. & Soliman, A.M. & Elwakil, A.S. & Sedeek, A., 2009. "On the stability of linear systems with fractional-order elements," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2317-2328.
    5. Khanzadeh, Alireza & Pourgholi, Mahdi, 2016. "Robust Synchronization of Fractional-Order Chaotic Systems at a Pre-Specified Time Using Sliding Mode Controller with Time-Varying Switching Surfaces," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 69-77.
    6. Martínez-Guerra, Rafael & Pérez-Pinacho, Claudia A. & Gómez-Cortés, Gian Carlo & Cruz-Victoria, Juan C., 2015. "Synchronization of incommensurate fractional order system," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 260-266.
    7. Chunlai Li & Jing Zhang, 2016. "Synchronisation of a fractional-order chaotic system using finite-time input-to-state stability," International Journal of Systems Science, Taylor & Francis Journals, vol. 47(10), pages 2440-2448, July.
    8. Peng, Qiu & Jian, Jigui, 2023. "Synchronization analysis of fractional-order inertial-type neural networks with time delays," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 62-77.
    9. Yi Chen & Jing Dong & Hao Ni, 2021. "ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 559-594, May.
    10. Pratap, A. & Raja, R. & Cao, J. & Lim, C.P. & Bagdasar, O., 2019. "Stability and pinning synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous activations," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 241-260.
    11. Li, Changpin & Yan, Jianping, 2007. "The synchronization of three fractional differential systems," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 751-757.
    12. Tam, Lap Mou & Si Tou, Wai Meng, 2008. "Parametric study of the fractional-order Chen–Lee system," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 817-826.
    13. G. Fern'andez-Anaya & L. A. Quezada-T'ellez & B. Nu~nez-Zavala & D. Brun-Battistini, 2019. "Katugampola Generalized Conformal Derivative Approach to Inada Conditions and Solow-Swan Economic Growth Model," Papers 1907.00130, arXiv.org.
    14. Gu, Haibo & Trujillo, Juan J., 2015. "Existence of mild solution for evolution equation with Hilfer fractional derivative," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 344-354.
    15. Lu, Jun Guo & Chen, Guanrong, 2006. "A note on the fractional-order Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 27(3), pages 685-688.
    16. Sheu, Long-Jye & Chen, Hsien-Keng & Chen, Juhn-Horng & Tam, Lap-Mou & Chen, Wen-Chin & Lin, Kuang-Tai & Kang, Yuan, 2008. "Chaos in the Newton–Leipnik system with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 98-103.
    17. Peng, Guojun & Jiang, Yaolin & Chen, Fang, 2008. "Generalized projective synchronization of fractional order chaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(14), pages 3738-3746.
    18. Laskin, Nick, 2018. "Valuing options in shot noise market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 502(C), pages 518-533.
    19. Shang, Li-Jen & Shyu, Kuo-Kai, 2009. "A method for extracting chaotic signal from noisy environment," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1120-1125.
    20. Ali Balcı, Mehmet, 2017. "Time fractional capital-induced labor migration model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 477(C), pages 91-98.

    More about this item

    Statistics

    Access and download statistics

    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:42:y:2009:i:2:p:1181-1189. 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.