Skip to main content
SpringerLink
Log in

Isochronicity and limit cycle oscillation in chemical systems

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

Chemical oscillation is an interesting nonlinear dynamical phenomenon which arises due to complex stability condition of the steady state of a reaction far away from equilibrium which is usually characterised by a periodic attractor or a limit cycle around an interior stationary point. In this context Lienard equation is specifically used in the study of nonlinear dynamical properties of an open system which can be utilized to obtain the condition of limit cycle. In conjunction with the property of limit cycle oscillation, here we have shown the condition for isochronicity for different chemical oscillators with the help of renormalisation group method with multiple time scale analysis from a Lienard system. When two variable open system of equations are transformed into a Lienard system of equation the condition for limit cycle and isochronicity can be stated in a unified way. For any such nonlinear oscillator we have shown the route of a dynamical transformation of a limit cycle oscillation to a periodic orbit of centre type depending on the parameters of the system.

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
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. A.H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981)

    Google Scholar 

  2. G.D. Birkhoff, Dynamical Systems (A.M.S. Publications, Providence, 1927)

    Book  Google Scholar 

  3. A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, Qualitative Theory of Second Order Dynamic Systems (Wiley, New York, 1973)

    Google Scholar 

  4. V.I. Arnold, Y. Ilyashenko, Ordinary Differential Equations, Encyclopedia Mathematical Science, 1st edn. (Springer, Berlin, 1988)

    Google Scholar 

  5. S. Smale, Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)

    Article  Google Scholar 

  6. D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)

    Article  Google Scholar 

  7. D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 23, 343–344 (1971)

    Article  Google Scholar 

  8. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer, New York, 1978)

    Google Scholar 

  9. K.G. Wilson, J. Kogut, Phase Transitions and Critical Phenomena, vol. 6, ed. by C. Domb, M.S. Green (Academic, New York) (1976)

  10. J. Guckenheimer, P. Holmes, Non-Linear Oscillations: Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1986)

    Google Scholar 

  11. L.Y. Chen, N. Goldenfeld, Y. Oono, Phys. Rev. Lett. 73, 1311 (1994)

    Article  CAS  Google Scholar 

  12. L.Y. Chen, N. Goldenfeld, Y. Oono, Phys. Rev. E 54, 376 (1996)

    Article  CAS  Google Scholar 

  13. D.W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations: An introduction for Scientists and Engineers, 4th edn. (Oxford University Press, Oxford, 2007)

    Google Scholar 

  14. A.A. Andronov, E.A. Leontovich, I.I. Gerdon, A.G. Maier, Qualitative Theory of Second Order Dynamic Systems (Wiley, New York, 1973)

    Google Scholar 

  15. S.N. Pandey, P.S. Bindu, M. Senthilvelan, M. Lakshmanan, J. Math. Phys. 50, 102701 (2009)

    Article  Google Scholar 

  16. A. Sarkar, J.K. Bhattacharjee, Eur. Phys. Letts. 91, 60004 (2010)

    Article  Google Scholar 

  17. A. Sarkar, J.K. Bhattacharjee, S. Chakraborty, D.B. Banerjee, Eur. Phys. J. D 64, 479 (2011)

    Article  CAS  Google Scholar 

  18. J.D. Murray, Non-Linear Differential Equation Models in Biology (Clarendon, Oxford, 1977)

    Google Scholar 

  19. J.D. Murray, Mathematical Biology (Springer, Berlin, 1989)

    Book  Google Scholar 

  20. J. Chavarriga, M. Sabatini, Qualitative Theory of Dynamical Systems 1, 1–70 (1999)

    Article  Google Scholar 

  21. S. Ghosh, D.S. Ray, Eur. Phys. J. B 87, 65 (2014)

    Article  Google Scholar 

  22. A. Babloyantz, A. Destexhe, Biol. Cybern. 58, 203–211 (1988)

    Article  CAS  Google Scholar 

  23. D. Pierson, F. Moss, Phys. Rev. Lett. 75, 21242127 (1995)

    Google Scholar 

  24. R. Refinetti, M. Menaker, Phys. Behav. 51(3), 613–637 (1992)

    Article  CAS  Google Scholar 

  25. M.E. Jewett, D.B. Forger, R.E. Kronauer, J. Biol. Rhythm. 14, 493 (1999)

    Article  CAS  Google Scholar 

  26. R.W. McCarley, W. Robert, S.G. Massaquoi, Am. J. Phys. Regul. Integr. Comp. Physiol. 251(6), R1011–R1029 (1986)

    CAS  Google Scholar 

  27. L. Glass, Nature 410(6825), 277–284 (2001)

    Article  CAS  Google Scholar 

  28. A. Goldbeter, Nature 420(6912), 238–245 (2002)

    Article  CAS  Google Scholar 

  29. G. Nicolis, I. Prigogine, Self-Organization in Non-equilibrium Systems (Wiley, New York, 1977)

    Google Scholar 

  30. J.F. Richard, R.M. Noyes, J. Chem. Phys. 60, 1877 (1974)

    Article  Google Scholar 

  31. J. Schnakenberg, J. Theor. Biol. 81, 389–400 (1979)

    Article  CAS  Google Scholar 

  32. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Dover Publications, New York, 2003)

    Google Scholar 

  33. A. Goldbeter, Biochemical Oscillations and Biological Rhythms (Cambridge University Press, Cambridge, 1996)

    Book  Google Scholar 

  34. I.R. Epstein, J.A. Pojman, An Introduction to Non-linear Chemical Dynamics (Oxford University Press, New York, 1998)

    Google Scholar 

  35. H.S. Strogatz, Non-linear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering (Westview Press, Boulder, 1994)

  36. A. Sarkar, P. Guha, A. Ghose-Choudhury, J.K. Bhattacharjee, A.K. Mallik, P.G.L. Leach, J. Phys. A Math. Theor. 45(41), 415101 (2012)

  37. P. Gray, S.K. Scott, Chemical Oscillations and Instabilities (Clarendon, Oxford, 1990)

    Google Scholar 

  38. W.C. Bray, J. Am. Chem. Soc. 43, 1262 (1921)

    Article  CAS  Google Scholar 

  39. W.C. Bray, H.A. Liebhafsky, J. Am. Chem. Soc. 53, 38 (1931)

    Article  CAS  Google Scholar 

  40. B.P. Belousov, Collection of Short Papers on Radiation Medicine (Medgiz, Moscow, 1958)

    Google Scholar 

  41. A.M. Zhabotinsky, Dokl. Akad. Nauk. USSR 157, 392 (1964)

    Google Scholar 

  42. R.J. Field, R.M. Noyes, J. Chem. Phys. 60, 1877 (1974)

    Article  CAS  Google Scholar 

  43. V. Beato, H. Engel, L. Schimansky-Geier, Eur. Phys. J. B 58, 323 (2007)

    Article  CAS  Google Scholar 

  44. E.E. Selkov, Eur. J. Biochem. 4, 79 (1968)

    Article  CAS  Google Scholar 

  45. J. Higgins, Proc. Natl. Acad. Sci. USA 51, 989 (1964)

    Article  CAS  Google Scholar 

  46. J.H. Merkin, D.J. Needham, S.K. Scott, Proc. Royal Soc. London A: Mathematical, Physical and Engineering Sciences 406, 299 (1986)

    Article  CAS  Google Scholar 

  47. S. Kar, D.S. Ray, Phys. Rev. Lett. 90, 238102 (2003)

    Article  Google Scholar 

  48. E.B. Postnikov, D.V. Verveyko, AYu. Verisokin, Phys. Rev. E 83, 062901 (2011)

    Article  CAS  Google Scholar 

  49. J.M.A.M. Kusters, J.M. Cortes, W.P.M. van Meerwijk, D.L. Ypey, A.P.R. Theuvenet, C.C.A.M. Gielen, Phys. Rev. Lett. 98, 098107 (2007)

    Article  CAS  Google Scholar 

  50. L. Chen, R. Wang, T.J. Kobayashi, K. Aihara, Phys. Rev. E 70, 011909 (2004)

    Article  Google Scholar 

  51. A. Goldbeter, Proc. R. Soc. Lond. B 261, 319 (1995)

    Article  CAS  Google Scholar 

  52. S. Sen, S.S. Riaz, D.S. Ray, J. Theor. Biol. 250, 103 (2008)

    Article  CAS  Google Scholar 

  53. A.I. Lavrova, L. Schimansky-Geier, E.B. Postnikov, Phys. Rev. E 79, 057102 (2009)

    Article  CAS  Google Scholar 

  54. A.I. Lavrova, E.B. Postnikov, Y.M. Romanovsky, Phys. Usp. 52, 1239 (2009)

    Article  CAS  Google Scholar 

Download references

Acknowledgements

Sandip Saha acknowledges RGNF, UGC, India for the partial financial support.

Author information

Authors and Affiliations

  1. S N Bose National Centre For Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata, 700106, India

    Sandip Saha & Gautam Gangopadhyay

Authors
  1. Sandip Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gautam Gangopadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gautam Gangopadhyay.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saha, S., Gangopadhyay, G. Isochronicity and limit cycle oscillation in chemical systems. J Math Chem 55, 887–910 (2017). https://doi.org/10.1007/s10910-016-0729-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-016-0729-1

Keywords

Search

Navigation