Skip to main content
SpringerLink
Log in

Stability analysis of Uçar prototype delayed system

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

A simple first-order differential equation involving delay proposed by Uçar is enriched with dynamical properties. The chaotic attractors are observed in this system for some values of delay. In this paper, we propose the stability results for this delayed system for arbitrary values of parameters using the method of critical curves. We discuss the effect of each parameter on stability and hence on the chaotic behavior. Our results are confirmed by the numerical observations available in the literature.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (2012)

    Google Scholar 

  2. Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York (1932)

    MATH  Google Scholar 

  3. May, R.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, NJ (1973)

    Google Scholar 

  4. Nisbet, R., Gurney, W.: Modelling Fluctuating Populations. Wiley, New York (1982)

    MATH  Google Scholar 

  5. Ruan, S.: Delay differential equations in single species dynamics. In: Arino, O., Hbid, M., Ait Dads, E. (eds.): Delay Differential Equations with Applications. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 205, Springer, Berlin p. 477 (2006)

  6. Kuang, Y. (ed.): Delay Differential Equations: With Applications in Population Dynamics. Academic Press, San Diego (1993)

    MATH  Google Scholar 

  7. Hethcote, H.W., Lewis, M.A., van den Driessche, P.: An epidemiological model with a delay and a nonlinear incidence rate. J. Math. Biol. 27, 49–64 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, New York (2001)

    Book  MATH  Google Scholar 

  9. Busenberg, S., Cooke, K.: Periodic solutions of a periodic nonlinear delay differential equation. SIAM J. Appl. Math. 35, 704–721 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. De Gaetano, A., Arino, O.: Mathematical modeling of intra-venous glucose tolerance test. J. Math. Biol. 40, 136–168 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mukhopadhyay, A., De Gaetano, A., Arino, O.: Modeling the intra-venous glucose tolerance test: a global study for a single-distributed delay model. Discrete Contin. Dyn. Syst.-B 4, 407–418 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Szydlowski, M., Krawiec, A.: Differential delay equations in chemical kinetics: some simple linear model systems. J. Chem. Phys. 92, 1702–1712 (1990)

    Article  Google Scholar 

  13. Szydlowski, M., Krawiec, A.: The Kaldor–Kalecki model of business cycles as a two-dimensional dynamical system. J. Nonlinear Math. Phys. 8, 266–271 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kyrychko, Y.N., Hogan, S.J.: On the use of delay equations in engineering applications. J. Vib. Control 16(78), 943–960 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Balachandran, B., Nagy, T.K., Gilsinn, D. (eds.): Delay Differential Equations: Recent Advances and New Directions. Springer, New York (2009)

    MATH  Google Scholar 

  16. Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feldstein, A.: Discretization methods for retarded ordinary differential equation. Ph.D. Thesis, Department of Mathematics, UCLA, Los Angeles (1964)

  18. Zwerkina, T.S.: Approximate solution of differential equation with retarded argument and differential equations with discontinuous right-hand side. Trudy Sem. Teor. Differential Uravnenii s Otklon. Argumemtom Univ. Druzby Narodov Patrisa Lumumby 1, 76–93 (1962)

  19. Zwerkina, T.S.: A modification of finite difference methods for integrating ordinary differential equations with non-smooth solutions. Z. Vycisl. Mat. Fiz. 4, 149–160 (1964)

    MathSciNet  Google Scholar 

  20. Zwerkina, T.S.: A modified Adams’ formula for the integration for equations with deviating argument. Trudy Sem. Teor. Differential Uravnenii s Otklon. Argumemtom Univ. Druzby Narodov Patrisa Lumumby 3, 221–232 (1965)

  21. Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, New York (2003)

    Book  MATH  Google Scholar 

  22. Bellman, R.: On the computational solution of differential-difference equations. J. Math. Anal. Appl. 2, 108–110 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  23. Evans, D.J., Raslan, K.R.: The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math. 00(0), 1–6 (2004)

  24. He, J.: Variational iteration method for delay differential equations. Commun. Nonlinear Sci. Numer. Simul. 2(4), 235–236 (1997)

    Article  Google Scholar 

  25. Shakeri, F., Dehghan, M.: Solution of delay differential equations via a homotopy perturbation method. Math. Comput. Model. 48(3–4), 486–498 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Uçar, A.: A prototype model for chaos studies. Int. J. Eng. Sci. 40, 251–258 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Uçar, A.: On the chaotic behaviour of a prototype delayed dynamical system. Chaos Solitons Fractals 16, 187–194 (2003)

    Article  MATH  Google Scholar 

  28. Strogatz, S.H.: Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Addison-Wesley, Massachusetts (1994)

    Google Scholar 

  29. Kan, M., Taguchi, H.: Ship capsizing and chaos. In: Thompson, J.M.T., Bishop, S.R. (eds.) Nonlinearity and Chaos in Engineering Dynamics, pp. 418–420. Wiley, Chichester (1994)

    Google Scholar 

  30. Lonngren, K.E., Bai, E.: On the Uçar prototype model. Int. J. Eng. Sci. 40, 1855–1857 (2002)

    Article  MATH  Google Scholar 

  31. Peng, M.: Bifurcation and chaotic behavior in the Euler method for a Uçar prototype delay model. Chaos Solitons Fractals 22, 483–493 (2004)

    Article  MATH  Google Scholar 

  32. Bai, E.W., Longren, K.E., Uçar, A.: Secure communication via multiple parameter modulation in a delayed chaotic system. Chaos Solitons Fractals 23(3), 1071–1076 (2005)

    Article  MATH  Google Scholar 

  33. Ghosh, D., Banerjee, S., Chowdhury, A.R.: Synchronization between variable time-delayed systems and cryptography. Europhys. Lett. 80(3), 30006 (2007)

    Article  MathSciNet  Google Scholar 

  34. Bhalekar, S.: Dynamical analysis of fractional order Uçar prototype delayed system. Signal Image Video Process. 6(3), 513–519 (2012)

    Article  Google Scholar 

  35. Bhalekar, S.: On the Uçar prototype model with incommensurate delays. Signal Image Video Process. 8(4), 635–639 (2014)

    Article  Google Scholar 

  36. Lakshmanan, M., Senthilkumar, D.V.: Dynamics of Nonlinear Time-Delay Systems. Springer, New York (2010)

    MATH  Google Scholar 

Download references

Acknowledgments

Author acknowledges the National Board for Higher Mathematics, Mumbai, India for the Research Grant (Ref. 2/48(6)/2013/NBHM(R.P.)/R&DII/689). Author also thanks anonymous referees for their insightful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur, 416004, India

    Sachin Bhalekar

Authors
  1. Sachin Bhalekar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sachin Bhalekar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhalekar, S. Stability analysis of Uçar prototype delayed system. SIViP 10, 777–781 (2016). https://doi.org/10.1007/s11760-015-0811-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-015-0811-3

Keywords

Search

Navigation