IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v121y2019icp30-38.html
   My bibliography  Save this article

Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller

Author

Listed:
  • Li, Wei
  • Huang, Dongmei
  • Zhang, Meiting
  • Trisovic, Natasa
  • Zhao, Junfeng

Abstract

Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker–Planck–Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods.

Suggested Citation

  • Li, Wei & Huang, Dongmei & Zhang, Meiting & Trisovic, Natasa & Zhao, Junfeng, 2019. "Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller," Chaos, Solitons & Fractals, Elsevier, vol. 121(C), pages 30-38.
    • Handle: RePEc:eee:chsofr:v:121:y:2019:i:c:p:30-38
    DOI: 10.1016/j.chaos.2019.01.026
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077919300360
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2019.01.026?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. Huang, Chengdai & Cao, Jinde & Xiao, Min & Alsaedi, Ahmed & Alsaadi, Fuad E., 2017. "Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 293-310.
    2. Li, Wei & Xu, Wei & Zhao, Junfeng & Jin, Yanfei, 2007. "Stochastic stability and bifurcation in a macroeconomic model," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 702-711.
    3. Lokenath Debnath, 2003. "Recent applications of fractional calculus to science and engineering," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-30, January.
    4. Kavyanpoor, Mobin & Shokrollahi, Saeed, 2017. "Challenge on solutions of fractional Van Der Pol oscillator by using the differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 44-45.
    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. Yong-Ge Yang & Ya-Hui Sun & Wei Xu, 2019. "Stochastic Bifurcations of a Fractional-Order Vibro-Impact System Driven by Additive and Multiplicative Gaussian White Noises," Complexity, Hindawi, vol. 2019, pages 1-10, October.

    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. Musrrat Ali & Hemant Gandhi & Amit Tomar & Dimple Singh, 2023. "Similarity Solution for a System of Fractional-Order Coupled Nonlinear Hirota Equations with Conservation Laws," Mathematics, MDPI, vol. 11(11), pages 1-14, May.
    2. 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.
    3. 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.
    4. Xing, Sheng Yan & Lu, Jun Guo, 2009. "Robust stability and stabilization of fractional-order linear systems with nonlinear uncertain parameters: An LMI approach," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1163-1169.
    5. Goharrizi, Amin Yazdanpanah & Khaki-Sedigh, Ali & Sepehri, Nariman, 2009. "Observer-based adaptive control of chaos in nonlinear discrete-time systems using time-delayed state feedback," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2448-2455.
    6. Liaqat, Muhammad Imran & Akgül, Ali, 2022. "A novel approach for solving linear and nonlinear time-fractional Schrödinger equations," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    7. Boukhouima, Adnane & Hattaf, Khalid & Lotfi, El Mehdi & Mahrouf, Marouane & Torres, Delfim F.M. & Yousfi, Noura, 2020. "Lyapunov functions for fractional-order systems in biology: Methods and applications," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    8. Ruiqing Shi & Ting Lu & Cuihong Wang, 2019. "Dynamic Analysis of a Fractional-Order Model for Hepatitis B Virus with Holling II Functional Response," Complexity, Hindawi, vol. 2019, pages 1-13, August.
    9. Laarem, Guessas, 2021. "A new 4-D hyper chaotic system generated from the 3-D Rösslor chaotic system, dynamical analysis, chaos stabilization via an optimized linear feedback control, it’s fractional order model and chaos sy," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    10. Ravi Kanth, A.S.V. & Devi, Sangeeta, 2021. "A practical numerical approach to solve a fractional Lotka–Volterra population model with non-singular and singular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    11. Jehad Alzabut & Weerawat Sudsutad & Zeynep Kayar & Hamid Baghani, 2019. "A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative," Mathematics, MDPI, vol. 7(8), pages 1-15, August.
    12. Iyiola, O.S. & Tasbozan, O. & Kurt, A. & Çenesiz, Y., 2017. "On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 94(C), pages 1-7.
    13. Ning, Xin & Ma, Yanyan & Li, Shuai & Zhang, Jingmin & Li, Yifei, 2018. "Response of non-linear oscillator driven by fractional derivative term under Gaussian white noise," Chaos, Solitons & Fractals, Elsevier, vol. 113(C), pages 102-107.
    14. Lashkarian, Elham & Reza Hejazi, S., 2017. "Group analysis of the time fractional generalized diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 479(C), pages 572-579.
    15. Sahadevan, R. & Prakash, P., 2017. "On Lie symmetry analysis and invariant subspace methods of coupled time fractional partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 107-120.
    16. Majumdar, Prahlad & Mondal, Bapin & Debnath, Surajit & Ghosh, Uttam, 2022. "Controlling of periodicity and chaos in a three dimensional prey predator model introducing the memory effect," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    17. Maneesha Gupta & Richa Yadav, 2013. "Optimization of Integer Order Integrators for Deriving Improved Models of Their Fractional Counterparts," Journal of Optimization, Hindawi, vol. 2013, pages 1-11, June.
    18. Erman, Sertaç & Demir, Ali, 2020. "On the construction and stability analysis of the solution of linear fractional differential equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    19. Serik Aitzhanov & Kymbat Bekenayeva & Zamira Abdikalikova, 2023. "Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator," Mathematics, MDPI, vol. 11(18), pages 1-12, September.
    20. Shuai Li & Chengdai Huang & Xinyu Song, 2019. "Bifurcation Based-Delay Feedback Control Strategy for a Fractional-Order Two-Prey One-Predator System," Complexity, Hindawi, vol. 2019, pages 1-13, April.

    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:121:y:2019:i:c:p:30-38. 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.