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Analysing the stability of a delay differential equation involving two delays

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Abstract

Analysis of systems involving delay is a popular topic among the applied scientists. In the present work, we analyse the generalised equation \(D^{\alpha } x(t) = g\left( x(t-\tau _1), x(t-\tau _2)\right) \) involving two delays, viz. \(\tau _1\ge 0\) and \(\tau _2\ge 0\). We use stability conditions to propose the critical values of delays. Using examples, we show that the chaotic oscillations are observed in the unstable region only. We also propose a numerical scheme to solve such equations.

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References

  1. I Podlubny, Fractional differential equations (Academic Press, New York, 1999)

    MATH  Google Scholar 

  2. S G Samko, A A Kilbas and O I Marichev, Fractional integrals and derivatives: Theory and applications (Gordon and Breach, Yverdon, 1993)

    MATH  Google Scholar 

  3. A A Kilbas, H M Srivastava and J J Trujillo, Theory and applications of fractional differential equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  4. F Mainardi, Fractional calculus and waves in linear viscoelasticity (Imperial College Press, London, 2010)

    Book  Google Scholar 

  5. R L Magin, Fractional calculus in bioengineering (Begll House Publishers, Danbury, 2006)

    Google Scholar 

  6. A Khan and A Tyagi, Pramana – J. Phys. 90: 67 (2018)

    Article  ADS  Google Scholar 

  7. V K Tamba, S T Kingni, G F Kuiate, H B Fotsin and P K Talla, Pramana – J. Phys. 91: 12 (2018)

    Article  ADS  Google Scholar 

  8. L Chen, Y He, X Lv and R Wu, Pramana – J. Phys. 85, 91 (2015)

    Article  ADS  Google Scholar 

  9. H Smith, An introduction to delay differential equations with applications to the life sciences (Springer, New York, 2010)

    Google Scholar 

  10. M Lakshmanan and D V Senthilkumar, Dynamics of nonlinear time-delay systems (Springer, Heidelberg, 2010)

    MATH  Google Scholar 

  11. C Lainscsek, P Rowat, L Schettino, D Lee, D Song, C Letellier and H Poizner, Chaos 22, 013119 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  12. C Lainscsek and T J Sejnowski, Chaos 23, 023132 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  13. D Matignon, IMACS, IEEE-SMC Proceedings on Computational Engineering in Systems and Application Multiconference (Lille, France, July 1996) Vol. 2, pp. 963–968

  14. M A Pakzad and S Pakzad, WSEAS Trans. Syst. 11, 541 (2012)

    Google Scholar 

  15. C Bonnet and J R Partington, Automatica 38, 1133 (2002)

    Article  Google Scholar 

  16. C Hwang and Y C Cheng, Automatica 42, 825 (2006)

    Article  Google Scholar 

  17. S Bhalekar, Pramana – J. Phys. 81, 215 (2013)

    Article  ADS  Google Scholar 

  18. S Bhalekar, Chaos 26, 084306 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  19. S Bhalekar and V Daftardar-Gejji, Commun. Nonlinear Sci. Numer. Simulat. 15, 2178 (2010)

    Article  ADS  Google Scholar 

  20. V Daftardar-Gejji, S Bhalekar and P Gade, Pramana – J. Phys. 79, 61 (2012)

    Article  ADS  Google Scholar 

  21. S Bhalekar, Signals Image Video Process. 6, 513 (2012)

    Article  Google Scholar 

  22. S Bhalekar, V Daftardar-Gejji, D Baleanu and R Magin, Comput. Math. Appl. 61, 1355 (2011)

    Article  MathSciNet  Google Scholar 

  23. S Bhalekar, V Daftardar-Gejji, D Baleanu and R Magin, Int. J. Bifurc. Chaos 22, 1250071 (2012)

    Article  Google Scholar 

  24. J K Hale and W Huang, J. Math. Anal. Appl. 178, 344 (1993)

    Article  MathSciNet  Google Scholar 

  25. J Belair and S A Campbell, SIAM J. Appl. Math. 54, 1402 (1994)

    Article  MathSciNet  Google Scholar 

  26. X Li and S Ruan, J. Math. Anal. Appl. 236, 254 (1999)

    Article  MathSciNet  Google Scholar 

  27. X P Wu and L Wang, J. Franklin Inst. 354, 1484 (2017)

    Article  MathSciNet  Google Scholar 

  28. P Bi and S Ruan, SIAM J. Appl. Dyn. Syst. 12, 1847 (2013)

    Article  MathSciNet  Google Scholar 

  29. S Gakkhar and A Singh, Commun. Nonlinear Sci. Numer. Simul. 17, 914 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  30. J K Hale and S M V Lunel, Introduction to functional differential equations (Springer-Verlag, New York, 1993)

    Book  Google Scholar 

  31. V Daftardar-Gejji, Y Sukale and S Bhalekar, Fract. Calc. Appl. Anal. 18, 400 (2015)

    Article  MathSciNet  Google Scholar 

  32. S Bhalekar, Signal Image Video Process. 8, 635 (2014)

    Article  Google Scholar 

  33. J G Lu, Chin. Phys. 15, 301 (2006)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author acknowledges the Science and Engineering Research Board (SERB), New Delhi, India, for the Research Grant (Ref. MTR / 2017 / 000068) under Mathematical Research Impact Centric Support (MATRICS) Scheme.

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  1. Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur, 416 004, India

    Sachin Bhalekar

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  1. Sachin Bhalekar
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Correspondence to Sachin Bhalekar.

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Bhalekar, S. Analysing the stability of a delay differential equation involving two delays. Pramana - J Phys 93, 24 (2019). https://doi.org/10.1007/s12043-019-1783-6

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  • DOI: https://doi.org/10.1007/s12043-019-1783-6

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