Skip to main content
SpringerLink
Log in

Bifurcation control in a delayed two-neuron fractional network

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

The issue of bifurcation control for a delayed fractional network involving two neurons is concerned. Delay-dependent stability conditions and the bifurcation point are established by discussing the associated characteristic equation of the proposed network. Then, a delayed feedback controller is firstly designed to stabilize the Hopf bifurcation, and desirable dynamics is achieved. It is indicated that the designed controller is extremely effective which can postpone the onset of bifurcation by carefully selecting the feedback gain. Finally, simulation results are given to verify the efficiency of the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Picozzi and B. West, “Fractional Langevin model of memory in financial markets,” Physical Review E, vol. 66, no. 4, pp. 46–118, 2002. [click]

    Article  MathSciNet  Google Scholar 

  2. E. Reyes-Melo, J. Martinez-Vega, C. Guerrero-Salazar and U. Ortiz-Mendez, “Application of fractional calculus to the modeling of dielectric relaxation phenomena in poly meric materials,” Journal of Applied Polymer Science, vol. 98, no. 2, pp. 923–935, 2005. [click]

    Article  Google Scholar 

  3. E. Ahmed and A. Elgazzar, “On fractional order differential equations model for nonlocal epidemics,” Physica A, vol. 379, no. 2, pp. 607–614, 2007. [click]

    Article  MathSciNet  Google Scholar 

  4. N. Özalp and E. Demirci, “A fractional order SEIR model with vertical transmission,” Mathematical and Computer Modelling, vol. 54, no. 1, pp. 1–6, 2011. [click]

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Song, S. Xu, and J. Yang, “Dynamical models of happiness with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, pp. 616–628, 2010. [click]

    Article  MathSciNet  MATH  Google Scholar 

  6. W. Deng, C. Li, and J. Lu, “Stability analysis of linear fractional differential system with multiple time delays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409–416, 2007. [click]

    Article  MathSciNet  MATH  Google Scholar 

  7. M. S. Tavazoei, M. Haeri, M. Siami, and S. Bolouki, “Maximum number of frequencies in oscillations generated by fractional order LTI systems,” IEEE Transaction on Signal Process, vol. 58, pp. 4003–4012, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. A. El-Saka, E. Ahmed, M. I. Shehata, and A. M. A. El-Sayed, “On stability, persistence, and Hopf bifurcation in fractional order dynamical systems,” Nonlinear Dynamics, vol. 56, pp. 121–126, 2009. [click]

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Y. T. Leung, H. Yang, and P. Zhu, “Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, pp. 1142–1155, 2014. [click]

    Article  MathSciNet  Google Scholar 

  10. J. Wang and X. Xiong, “A general fractional-order dynamical network: synchronization behavior and state tuning,” Chaos, vol. 22, art. 023102, 2012.

  11. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990. [click]

    Article  MathSciNet  MATH  Google Scholar 

  12. C. K. Ahn, P. Shi, and M. V. Basin, “Two-dimensional dissipative control and filtering for roesser model,” IEEE Transaction on Automatic Control, vol. 60, no. 7, pp. 1745–1759, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. B. Li, P. Shi, L. G. Wu, and X. Zhang, “Fuzzy-modelbasedD-stability and nonfragile control for discrete-time descriptor systems with multiple delays,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 4, pp. 1019–1025, 2014.

    Article  Google Scholar 

  14. G. Chen, “On some controllability conditions for chaotic dynamics control,” Chaos, Solitons and Fractals, vol. 8, no. 9, pp. 1461–1470, 1997. [click]

    Article  MathSciNet  Google Scholar 

  15. M. Zhang, X. N. Shen, and T. Li, “Fault tolerant attitude control for cube sats with input saturation based on dynamic adaptive neural network,” International Journal of Innovative Computing, Information and Control, vol. 12, no. 2, pp. 651–663, 2016.

    Google Scholar 

  16. F. Padula, S. Alćl ćntara, R. Vilanova, and A. Visioli, “H ∞ control of fractional linear systems,” Automatica, vol. 49, no. 7, pp. 2276–2280, 2013. [click]

    Article  MathSciNet  Google Scholar 

  17. M. S. Abdelouahab, N. E. Hamri, and J. W. Wang, “Hopf bifurcation and chaos in fractional-order modified hybrid optical system,” Nonlinear Dynamics, vol. 69, no. 1, pp. 279–284, 2012.

    MathSciNet  MATH  Google Scholar 

  18. X. Li and R. C. Wu, “Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system,” Nonlinear Dynamics, vol. 78, no. 1, pp. 279–288, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Xiao, W. X. Zheng, G. P. Jiang and J. D. Cao, “Undamped oscillations generated by hopf bifurcations in fractional-order recurrent neural networks with caputo derivative,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 12, pp. 3201–3214, 2015.

    Article  MathSciNet  Google Scholar 

  20. M. Xiao, G. P. Jiang, W. X. Zheng, S. L. Yan, Y. H. Wan, and C. X. Fan, “Bifurcation control of a fractional-order van der pol oscillator based on the state feedback,” Asian Journal of Control, vol. 17, no. 5, pp. 1756–1766, 2015. [click]

    Article  MathSciNet  MATH  Google Scholar 

  21. C. D. Huang, J. D. Cao, and M. Xiao, “Hybrid control on bifurcation for a delayed fractional gene regulatory network,” Chaos, Solitons and Fractals, vol. 87, pp. 19–29, 2016. [click]

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Shi and Z. H. Wang, “Stability and hopf bifurcation control of a fractional-order small world network model,” Science China Physics, Mechanics and Astronomy, vol. 43, no. 4, pp. 467–477, 2013.

    Article  Google Scholar 

  23. M. Xiao, D. W. C. Ho, and J. D. Cao, “Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation,” Nonlinear Dynamics, vol. 58, pp. 319–344, 2009. [click]

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Shi, F. Li, L. G. Wu, and C. C. Lim, “Neural network-based passive filtering for delayed neutraltype semi-markovian jump systems,” IEEE Transactions on Neural Networks and Learning Systems, DOI: 10.1109/TNNLS.2016.2573853.

  25. H. Matsuura, “Probability of fuzzy set theory and probability amplitude of quantum neurons,” International Journal of Innovative Computing, Information and Control, vol. 12, no. 2, pp. 503–518, 2016.

    Google Scholar 

  26. J. D. Cao and M. Xiao, “Stability and Hopf bifurcation in a simplified BAM neural network with two time delays,” IEEE Transaction on Neural Networks, vol. 18, pp. 416–30, 2007.

    Article  Google Scholar 

  27. C. K. Ahn, P. Shi, and L. G. Wu, “Receding horizon stabilization and disturbance attenuation for neural networks With time-varying delay,” IEEE Transaction on Cybernetics, vol. 45, no. 12, pp. 2680–2692, 2015.

    Article  Google Scholar 

  28. F. B. Biao, L. G. Wu, and P. Shi, “Stochastic stability of semi-Markovian jump systems with mode-dependent delays,” International Journal of Robust and Nonlinear Control, vol. 24, no. 18, pp. 3317–3330, 2014. [click]

    Article  MathSciNet  MATH  Google Scholar 

  29. B. Lundstrom, M. Higgs, W. Spain, and A. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, pp. 1335–1342, 2008. [click]

    Article  Google Scholar 

  30. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

    MATH  Google Scholar 

  31. J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter, Berlin, 2001.

    Book  MATH  Google Scholar 

  32. C. D. Huang, J. D. Cao, M. Xiao, A. Alsaedi, and T. Hayat, “Bifurcations in a delayed fractional complex-valued neural network”,Applied Mathematics and Computation, vol. 292, pp.210–227, 2017.

    Article  MathSciNet  Google Scholar 

  33. C. X. Huang and L. H. Huang, “Existence and global exponential stability of periodic solutions of two-neuron networks with time-varying delays,” Applied Mathematics Letters, vol. 19, pp. 126–134, 2006. [click]

    Article  MathSciNet  MATH  Google Scholar 

  34. L. Olien and J. Bélair, “Bifurcations, stability, and monotonicity properties of a delayed neural network model,” Physica D, vol. 102, pp. 349–363, 1997. [click]

    Article  MathSciNet  MATH  Google Scholar 

  35. E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz,” Rossler, Chua and Chen Systems, Physics Letter A, vol. 358. pp. 1–4, 2006.

    MATH  Google Scholar 

  36. M. Xiao, W. X. Zheng, and J. D. Cao, “Frequency domain approach to computational analysis of bifurcation and periodic solution in a two-neuron network model with distributed delays and self-feedbacks,” Neurocomputing, vol. 99, pp. 206–213, 2013. [click]

    Article  Google Scholar 

  37. C. X. Huang, L. H. Huang, J. F. Feng, M. Y. Nai, and Y. G. He, “Hopf bifurcation analysis for a two-neuron network with four delays,” Chaos, Solitions and Fractals, vol. 34, pp. 795–812, 2007. [click]

    Article  MathSciNet  MATH  Google Scholar 

  38. C. K. Ahn, L. G. Wu, and P. Shi, “Stochastic stability analysis for 2-D Roesser systems with multiplicative noise,” Automatica, vol. 69, pp. 356–363, 2016. [click]

    Article  MathSciNet  MATH  Google Scholar 

  39. C. K. Ahn, P. Shi, and M. V. Basin, “Deadbeat dissipative FIR filtering,” IEEE Trans. on Circuits and Systems I, vol. 63, no. 8, pp.1210–1221, 2016.

    Article  MathSciNet  Google Scholar 

  40. S. Bhalekar and D. Varsha, “A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order,” International Journal of Fractional Calculus and Application, vol. 1, no. 5, pp. 1–9, 2011.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Center for Complex Systems and Network Sciences, and School of Mathematics, Southeast University, Nanjing, 210096, China

    Lingzhi Zhao, Jinde Cao & Chengdai Huang

  2. School of Information Engineering, Nanjing Xiaozhuang University, Nanjing, 211171, China

    Lingzhi Zhao

  3. Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Jinde Cao & Ahmed Alsaedi

  4. Information Systems Department, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Abdullah Al-Barakati & Habib M. Fardoun

Authors
  1. Lingzhi Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinde Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chengdai Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ahmed Alsaedi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Abdullah Al-Barakati
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Habib M. Fardoun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinde Cao.

Additional information

Recommended by Associate Editor Choon Ki Ahn under the direction of Editor Hamid Reza Karimi. This work was jointly supported by the National Natural Science Foundation of China under Grant nos. 61573096 and 61272530, the National Science Foundational of Jiangsu Province of China under Grant no. BK2012741, the “333 Engineering” Foundation of Jiangsu Province of China under Grant no. BRA2015286.

Lingzhi Zhao received her B.S. degree in Mathematics/Applied Mathematics in 2003 from Qufu Normal University, Qufu, China, and her M.S. degree in applied mathematics in 2006 from Shanghai University, Shanghai, China. Currently she is a lecturer with the School of Information Engineering, Nanjing Xiaozhuang University, Nanjing, China. Her current research interests include fractional calculus theory, bifurcation control theory, neural networks, and complex dynamical networks.

Jinde Cao is a Distinguished Professor, the Dean of School of Mathematics and the Director of the Research Center for Complex Systems and Network Sciences at Southeast University. From March 1989 to May 2000, he was with the Yunnan University. In May 2000, he joined the School of Mathematics, Southeast University, Nanjing, China. From July 2001 to June 2002, he was a Postdoctoral Research Fellow at the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. Professor Cao was an Associate Editor of the IEEE Transactions on Neural Networks, Journal of the Franklin Institute, Neurocomputing, and Differential Equations and Dynamical Systems. He is currently an Associate Editor of the IEEE Transactions on Cybernetics, IEEE Transactions on Cognitive and Developmental Systems, Neural Networks, Nonlinear Analysis: Modelling and Control, Mathematics and Computers in Simulation, Advances in Difference Equations, and Journal of Artificial Intelligence and Soft Computing Research. He has been named as Highly-Cited Researcher in Mathematics, Computer Science and Engineering by Thomson Reuters.

Chengdai Huang received his B.S. degree in mathematics/applied mathematics in 2002 from Hunan University of Science and Technology, Xiangtan, China, and his M.S. degree in applied mathematics in 2013 from Nanchang Hangkong University, Nanchang, China. Since September 2013, he has been pursuing his Ph.D. degree in applied mathematics with the School of Mathematics, Southeast University, Nanjing, China. His current research interests include complex networks, bifurcation control, synchronization and fractional-order systems

Ahmed Alsaedi obtained his Ph.D. degree from Swansea University (UK) in 2002. He has a broad experience of research in applied mathematics. His fields of interest include dynamical systems, nonlinear analysis involving ordinary differential equations, fractional differential equations, boundary value problems, mathematical modeling, biomathematics, Newtonian and Non-Newtonian fluid mechanics. He has published several articles in peer-reviewed journals. He has supervised several M.S. students and executed many research projects successfully. He is reviewer of several international journals. He served as the chairman of the mathematics department at KAU and presently he is serving as director of the research program at KAU. Under his great leadership, this program is running quite successfully and it has attracted a large number of highly rated researchers and distinguished professors from all over the world. He is also the head of NAAM international research group at KAU.

Abdullah Al-Barakati is the general supervisor of the IT department of Makkah Province in Saudi Arabia. He is also an assistant Professor at King Abdulaziz University, Saudi Arabia. Prior to that, he was the head of the Information Systems Department at King Abdulaziz University, Saudi Arabia. He received a BSc (Hons) in Computer Science in 2004, MSc in Software Engineering, a PhD in Computer Science in 2012 from University of Sussex, United Kingdom. His research interests revolve on the utilization Big Data and Web technologies for innovative solutions and services. He has published more than 21 peer reviewed research paper on Workflow Systems, Social Media, Big Data and Digital Heritage.

Habib M. Fardoun has a PhD in Model- Based Approach for the Development of Quality Higher Educational Environments (2011), and Master in Aptitude for Teaching (2008), and Master in Advanced Computer Technologies (2007), by the University of Castilla-La Mancha. Currently, he is occupying the position of Associate Professor at the Faculty of Computing and Information Technology (Jeddah, Saudi Arabia). Habib has more than 100 publications (between journals, chapter books and conferences) in the field of Human-Computer Interaction (HCI), Education and Rehabilitation. He served as co-chair of more than 10 conferences and workshops, and as guest editor of more than 15 special issue of ISI journals.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, L., Cao, J., Huang, C. et al. Bifurcation control in a delayed two-neuron fractional network. Int. J. Control Autom. Syst. 15, 1134–1144 (2017). https://doi.org/10.1007/s12555-016-1271-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-016-1271-4

Keywords

Search

Navigation