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Anti-control of Hopf bifurcation for the Willamowski–Rössler system

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Abstract

Anti-control of Hopf bifurcation is one of the hot topics in nonlinear dynamics research, which is to make the system generate or strengthen bifurcation at a prespecified location. The Willamowski–Rössler system is taken as the research object, which is a nonlinear dynamic system derived from chemical reaction processes. Using the higher-dimensional Hopf bifurcation theory, the critical value of the Hopf bifurcation of a non-zero equilibrium point is obtained. A state feedback control method is proposed. With this method, anti-control of Hopf bifurcation for the system is accomplished by a hybrid controller, which is composed of linear and nonlinear controllers. The relationship between the bifurcation parameter and the control parameters of the linear controller is obtained. The values of the control parameters of the linear controller are determined by the bifurcation parameter. Although the critical values of the bifurcation parameters are not determined by the control parameters of the nonlinear controller, the parameter can change the amplitude of the limit cycle, which is inversely proportional to the amplitude of the limit cycle. Finally, the theoretical analysis is verified by numerical simulation.

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References

  1. Zhang L, Tang JS, Han Q (2018) Hopf bifurcation control of a Pan-Like chaotic system. Chin Phys B 27(9):094702

    Article  Google Scholar 

  2. Zhu LH, Zhao HY, Wang XM (2015) Bifurcation analysis of a delay reaction–diffusion malware propagation model with feedback control. Commun Nonlinear Sci Numer Simul 22(1):747–768

    Article  MathSciNet  Google Scholar 

  3. Xu Y, Ma SJ, Zhang HQ (2011) Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method. Nonlinear Dyn 65(1):77–84

    Article  MathSciNet  Google Scholar 

  4. Ding DW, Wang C, Ding LH et al (2016) Hopf bifurcation control in a FAST TCP and RED model via multiple control schemes. J Control Sci Eng 2016(4):1–10

    Article  MathSciNet  Google Scholar 

  5. Ding YT, Jiang WH, Wang HB (2010) Delayed feedback control and bifurcation analysis of Rossler chaotic system. Nonlinear Dyn 61(4):707–715

    Article  MathSciNet  Google Scholar 

  6. Cheng ZS (2010) Anti-control of Hopf bifurcation for Chen’s system through washout filters. Neurocomputing 73(16):3139–3146

    Article  Google Scholar 

  7. Yang Y, Liao XF, Dong T (2017) Anti-control of Hopf bifurcation in the Shimizu-Morioka system using an explicit criterion. Nonlinear Dyn 89(1):1453–1461

    Article  Google Scholar 

  8. Chen DS, Wang HO, Chen GR (2001) Anti-control of Hopf bifurcations. IEEE Trans Circuits Syst IFTA 48(6):661–672

    Article  MathSciNet  Google Scholar 

  9. Liu SH, Tang JS (2008) Anti-control of Hopf bifurcation at zero equilibrium of 4D Qi system. Acta Phys Sin 57(10):6162–6167

    Article  MathSciNet  Google Scholar 

  10. Zhang SJ, Du WX, Yin S (2017) Anti-control of Double Hopf bifurcation of vibration rating griddle system. J Hunan Univ (Nat Sci) 44(10):55–61

    Google Scholar 

  11. Wen GL, Xu HD, Lv ZY et al (2015) Anti-controlling Hopf bifurcation in a type of centrifugal governor system. Nonlinear Dyn 81(1):811–822

    Article  Google Scholar 

  12. Bodalea ILIE (2015) VICTOR Andrei Oancea. Chaos control for Willamowski–Rössler model of chemical reactions. Chaos Solitons Fractals 78:1–9

    Article  MathSciNet  Google Scholar 

  13. Aguda BD, Clarke BL (1988) Dynamic elements of chaos in the Wiliamowski–Rössler network. J Chem Phys 89:7428

    Article  MathSciNet  Google Scholar 

  14. Sun WP, Wang HY, Kan M (2018) Numerical simulation, control and synchronization research on chaotic behavior of Wiliamowski–Rössler System. J Dyn Control 16(1):35–40

    Google Scholar 

  15. Wu XG, Raymond K (1993) Internal fluctuations and deterministic chemical chaos. Phys Rev Lett 70(13):1940–1943

    Article  Google Scholar 

  16. Chávez F, Kapral R (2002) Oscillatory and chaotic dynamics in compartmentalized geometries. Phys Rev 65:056203

    Google Scholar 

  17. Wang HL, Xin HW (1995) Control of chemical chaos and influence of fluctuations. Chin J Chem Phys 8(2):129–132

    Google Scholar 

  18. Stucki JW, Urbanczik R (2005) Entropy production of the Willamowski–Rössler Oscillator. Zeitschrift Für Naturforschung A 60a:599–607

    Article  Google Scholar 

  19. Nguyen LH, Hong KS (2012) Hopf bifurcation control via a dynamic state-feedback control. Phys Lett A 376:442–446

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the financial support received by the National Natural Science Foundation of China (No. 11372102), the PhD Start-up Fund of Jianghan University under Grant (No. 1029/06040001),and the Guiding Project of Science and Technology Research Plan of Hubei Provincial Department of Education (No. B2022458).

Funding

The National Natural Science Foundation of China (No. 11372102), the PhD Start-up Fund of Jianghan University under Grant(No. 1029/06040001), and the Guiding Project of Science and Technology Research Plan of Hubei Provincial Department of Education (No. B2022458).

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Authors and Affiliations

  1. School of Intelligent Manufacturing, Jianghan University, Wuhan, 430056, People’s Republic of China

    Liang Zhang, Ziqiang Fang & Songlin Peng

  2. College of Mechanical and Electrical Engineering, Wuhan Qingchuan University, Wuhan, 430204, People’s Republic of China

    Qin Han

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  1. Liang Zhang
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  2. Qin Han
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Contributions

Liang Zhang is mainly responsible for analyzing and writing. Qin Han is responsible for partial simulation analysis. Ziqiang Fang is responsible for partial simulation analysis. Songlin Peng is responsibal for revising this paper.

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Correspondence to Liang Zhang.

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Zhang, L., Han, Q., Fang, Z. et al. Anti-control of Hopf bifurcation for the Willamowski–Rössler system. Int. J. Dynam. Control 12, 1562–1570 (2024). https://doi.org/10.1007/s40435-023-01264-9

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  • DOI: https://doi.org/10.1007/s40435-023-01264-9

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