[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Numerical Study on Bray-Liebhafsky Oscillatory Reaction: Bifurcations

Journal of Applied Nonlinear Dynamics 2(3) (2013) 285--301 | DOI:10.5890/JAND.2013.08.004

Branislav Stanković$^{1}$; Željko Čupić$^{2}$; Nataša Pejić$^{3}$; Ljiljana Kolar-Anić$^{1}$

$^{1}$ Faculty of Physical Chemistry, University of Belgrade, Studentski trg 12-16, 11000 Belgrade, Serbia

$^{2}$ Institute of Chemistry, Technology and Metallurgy, University of Belgrade, Department of Catalysis and Chemical Engineering, Njegoševa 12, 11000 Belgrade, Serbia

$^{3}$ Faculty of Pharmacy, University of Belgrade, Vojvode Stepe 450, 11000 Belgrade, Serbia

Download Full Text PDF

 

Abstract

The time series obtained by numerical simulations of the model of the Bray-Liebhafsky oscillatory reaction is analyzed under the continuously fed well stirred tank reactor (CSTR) conditions, with the aim to find bifurcation points in which system of Bray-Liebhafsky oscillatory reaction transforms from stable to unsta- ble state and vice versa. Types of bifurcation points, supercrit- ical and subcritical Andronov-Hopf bifurcation and saddle-loop bifurcation are determined from characteristic scaling laws.

Acknowledgments

This investigation was supported by the Ministry of Education and Science of Republic of Serbia, under the projects No. 172015 and 45001.

References

  1. [1]  Miller, W.L. and Chrousos G.P. (2001), The Adrenal Cortex, in Felig P., Frohman L. (eds.) Endocrinology & Metabolism, McGraw-Hill, New York, 387-524.
  2. [2]  Hartmann, A., Veldhuis, J.D., Deuschle, M., Standhardt, H., and Heuser, I. (1997), Twenty-four hour cortisol release profiles in patients with Alzheimer's and Parkinson's disease compared to normal controls: ultradian secretory pulsatility and diurnal variation, Neurobiology of Aging, 18, 285-289.
  3. [3]  Henley, D.E., Leendertz, J.A. , Russell, G.M., Wood, S.A., Taheri, S., Woltersdorf W.W., and Lightman S.L. (2009), Development of an automated blood sampling system for use in humans, Journal of Medical Engineering & Technology, 33, 199-208.
  4. [4]  Jelić, S., Čupić, Ž., and Kolar-Anić Lj. (2005), Mathematical modeling of the hypothalamic-pituitaryadrenal system activity, Mathematical Biosciences, 197, 173-187.
  5. [5]  Jelić, S., Čupić, Ž., and Kolar-Anić Lj., and Vukojević, V. (2009), Predictive modeling of the hypothalamicpituitary- adrenal (HPA) function. Dynamic systems theory approach stoichiometric network analysis and quenching small amplitude oscillation, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1451-1472.
  6. [6]  Marković, V., Čupić, Ž., Vukojević, V., and Kolar-Anić Lj. (2011), Predictive modeling of the hypothalamic-pituitary-adrenal (HPA) axis response to acute and chronic stress, ENDOCR J, 58, 889- 904.
  7. [7]  Marković, V.M. Čupić, Ž., Ivanovć A., and Kolar-Anić, Lj. (2011), The Stability of the Extended Model of Hypothalamic-Pituitary-Adrenal (HPA) Axis examined by Stoichiometric Network Analysis (SNA),Russian Journal of Physical Chemistry A, 85, 2327-2335.
  8. [8]  Duysens, L.N.M. and Amesz, J. (1957), Fluorescence Spectrophotometry of Reduced Phosphopyridine Nucleotide in Intract Cell in the Near-Ultraviolet and Visible region, Biochimica et Biophysica Acta, 24, 19-26.
  9. [9]  Gosh, A. and Chance, B. (1964), Oscillations of Glycolic Intermediates in Yeast Cells, Biochemical and Biophysical Research Communications, 16, 174-181.
  10. [10]  Chance, B., Eastbrook, W., and Gosh, A. (1964), Damped Sinusoidal Oscillations of Cytoplasmic Reduced Pyridine Nucleotide in Yeast Cells, Proceedings of the National Academy of Sciences USA, 51, 1244-1251.
  11. [11]  Bray, W.C. (1921), Periodic reaction in homogenous solution and its relation to catalysis, Journal of the American Chemical Society, 43, 1262-1261.
  12. [12]  Bray, W.C. and Liebhafsky, H.A. (1931), Reaction involving hydrogen peroxide, iodine and iodate ion. I. Introduction, Journal of the American Chemical Society, 53, 38-44.
  13. [13]  Liebhafsky, H.A., McGavock, W.C., Reyes, R.J., Roe, G.M., and Wu, L.S. (1978), Reaction involving hydrogen peroxide, iodine, and iodate ion. 6. Oxidation of iodine by hydrogen peroxide at 50°C, Journal of the American Chemical Society, 100, 87-91.
  14. [14]  Schmitz, G. (1987), Cinetique de la reaction de Bray, Journal of Chemical Physics, 84, 957-965.
  15. [15]  Anić, S. and Kolar-Anić, Lj. (1988), Kinetic aspect of the Bray-Liebhafsky oscillatory reaction, Journal of the Chemical Society, Faraday Transactions I, 84, 3413-3421.
  16. [16]  Anić, S., Kolar-Anić, Lj., Stanisavljev, D., Begović, N., and Mitić, D. (1991), Dilution reinitiated oscillations in the Bray-Liebhafsky system, Reaction Kinetics and Catalysis Letters, 43, 155-162.
  17. [17]  Belousov, B.P. (1958), A periodic reactions and its mechanism, Sbornik Referatov po Radiatsionnoi Meditsine, Medigz, Moskow, pp. 145.
  18. [18]  Zhabotinskii, A.M. (1964), Periodic processes of the oxidation of malonic acid in solution, Biofizika, 9, 306-311.
  19. [19]  Zhabotinskii, A.M. and Frank, G. (1967), Oscillating Processes in Biological and Chemical System, Nauka, Moskow.
  20. [20]  Zhabotinskii, A.M. (1974), Koncentrationnye Avtokolebanya, Nauka, Moskva. (in russian).
  21. [21]  Körös, E., Burger, M., Friedrich, V., Ladanyi, L., Nagy, Z., and Orban, M. (1974), Chemistry of Belousov Type Oscillating Reactions, Faraday Symposia of the Chemical Society, 9, 28-37.
  22. [22]  Stanisavljev, D, Begović, N., and Vukojević, V. (1998), Influence of Heavy Water on the Bray-Liebhafsky Oscillating Reaction, Journal of Chemical Physics A, 102, 6887-6891.
  23. [23]  Blagojević, S. M., Ani, S. R., Čupić, Ž. D. Pejić, N.D., and Kolar-Anić, Lj.Z. (2008), Malonic acid concentration as a control parameter in the kinetic analysis of the Belousov-Zhabotinsky reaction under batch conditions, Physical Chemistry Chemical Physics, 10, 6658-6664.
  24. [24]  Briggs, T.S. and Rausher, W.C. (1973), An oscillatory iodine clock, Journal of Chemical Education, 50, 496-499.
  25. [25]  Furrow, S.D. (1981), Briggs-Rauscher oscillator with methylmalonic acid, Journal of Chemical Physics, 85, 2026-2031.
  26. [26]  Anić, S., Vukojević, V., Radenković, M., and Kolar-Anić, Lj. (1989), New approach to the study of the peroxide kinetics of the Briggs-Rausher oscillatory reaction, Journal of the Serbian Chemical Society, 54, 521-562.
  27. [27]  Vukojević, V., Graae Sorensen, P., and Hynne, F. (1993), Quenching analysis of the Briggs-Rauscher reaction, Journal of the American Chemical Society, 97, 4091-4100.
  28. [28]  Furrow, S. D., Cervellati, R., and Amadori, G. (2002), New substrates for the oscillating Briggs Rauscher reaction, Journal of Chemical Physics A, 106, 5841-5850.
  29. [29]  Orban, M., Körös, E, (1978), Chemical Oscillations During the Uncatalyzed Reaction of Aromatic Compounds with Bromate. Part 1. Search for Chemical Oscillators, Journal of Chemical Physics, 82(14), 1672-1674.
  30. [30]  Field, R.J. and Burger, M. (Eds.), (1985), Oscillations and Traveling Waves in Chemical Systems, Wiley, New York.
  31. [31]  Gray, P. and Scott, S.K. (1990), Chemical Oscillations and Instabilities, Clarendon Press, Oxford.
  32. [32]  Gray, P., Nicolis, G.,Baras, F., Borckmans, P., and Scott, S.K. (1990)(Eds.), Spatial Inhomogeneities and Transient Behaviour in Chemical Kinetics, Manchester University Press, Manchester.
  33. [33]  De Brouwer, S., Edwards, D.H., and Griffith, T.M. (1998), Simplification of the quasiperiodic route to chaos in agonist-induced vasomotion by iterative circle maps, American Journal of Physiology. Heart and Circulatory Physiology, 274, H1315-H1326.
  34. [34]  Griffith, T.M. and Edwards, D.H. (1995), Complexity of chaotic vasomotion is insensitive to flow and pressure but can be regulated by external control, American Journal of Physiology. Heart and Circulatory Physiology, 269, H656-H668.
  35. [35]  Braun, T., Lisboa, J.A. and Gallas, J.A.C. (1992), Evidence of homoclinic chaos in the plasma of glow discharge, Physical Review Letters, 68, 2770-2773.
  36. [36]  Hayashi, T. (2000), Mixed-mode oscillations and chaos in a glow discharge, Physical Review Letters, 84, 3334-3337.
  37. [37]  Varma, R.K., Shukla, P.K., and Krishan, V. (1993), Electrostatic oscillations in the presence of graincharge perturbations in dusty plasmas, Physical Review E, 47, 3612-3616.
  38. [38]  Mikikian, M., Cavarroc, M., Couëdel, L., Tessier, Y., and Boufendi L. (2008), Mixed-Mode Oscillations in Complex Plasma Instabilities, Physical Review Letters, 100, 225005-225008.
  39. [39]  Hudson, J.L. and Tsotsis, T.T. (1994), Eletrochemical reaction dynamics: a rewiew, Chemical Engineering Science, 49, 1493-1572.
  40. [40]  Blackburn, J.A., Smith, H.J.T., and Grønbech-Jensen, N. (1992), Stability and Hopf bifurcation in an inverted pendulum, American Journal of Physics, 60(10), 903-908.
  41. [41]  Zakrzhevsky, M., Klokov, A., Yevstignejev, V., and Shilvan, E. (2011), Complete bifurcation analysis of driven damped pendulum system, Estonian Journal of Engineering, 17(1), 76-87.
  42. [42]  Luo, A.C.J. and Huang, J. (2012), Analitical Dynamics of period-m flows and chaos in nonlinear systems, International Journal of Bifurcationand Chaos, 22, Article No: 1250093 (29 pages).
  43. [43]  Edelson, D. and Noyes, R.M. (1979), Detailed Calculations Modeling the Oscillatory Bray-Liebhafsky Reaction, Journal of Chemical Physics, 83, 212-220.
  44. [44]  Kolar-Anić, Lj. and Schmitz, G. (1992), Mechanism of the Bray-Liebhafsky reaction: Effect of the oxidation of iodous acid by hydrogen peroxide, Journal of the Chemical Society, Faraday Transactions, 88, 2343-2349.
  45. [45]  Kolar-Anić, Lj., Mišljenović, D., Anić, S. and Nicolis, G. (1995), Influence of the reduction of iodate by hydrogen peroxide on the model of the Bray-Liebhafsky reaction, Reaction Kinetics and Catalysis Letters, 54, 35-41.
  46. [46]  Valent, I., Adamčikova, L., and Ševčik, P. (1998), Simulations of the iodine interphase transport effect on the oscillating Bray-Liebhafsky reaction, Journal of Chemical Physics A, 102, 7576-7579.
  47. [47]  Kolar-Anić, Lj., Čupić, Ž., Schmitz, G. and Anić, S. (2010), Improvement of the stoichiometric network analysis for determination of instability conditions of complex nonlinear reaction systems, Chemical Engineering Science, 65, 3718-3728.
  48. [48]  Schmitz, G., Kolar-Anić, Lj., Anić, S., Grozdić, T. and Vukojević, V. (2006), Complex and Chaotic Oscillations in a Model for the Chatalytic Hydrogen Peroxide Decomposition under Open Reactor Conditions, Journal of Chemical Physics A, 110, 10361-10368.
  49. [49]  Ivanović, A.Z., Čupić, Ž. D, Kolar-Anić, Lj. Z., Janković, M.M. and Anić, S.R. (2009), Large Deviation Spectra of Chaotic Time Series from Bray-Liebhafsky Reaction, Russian Journal of Physical Chemistry A, 83(9), 1526-1530.
  50. [50]  Ivanović-Šaši′c, A., Marković, V., Anić, S., Kolar-Anić, Lj., and Čupić, Ž. (2011), Structures of chaos in open reaction systems, Physical Chemistry Chemistry Physical, 13, 20162-20171.
  51. [51]  Čupić, Ž., Ivanović-Šaši′c, A., S. Anić, Stanković, B., Maksimović, J., Kolar-Anić, Lj., and Schmitz, G. (2013), Tourbillion in the Phase Space of the Bray-Liebhafsky Nonlinear Oscillatory Reaction and Related Multiple-Time-Scale Model, MATCH: Communications in Mathematical and in Computer Chemistry, 69, 805-830.
  52. [52]  Kolar-Anić, Lj. and Anić, S. (1997), Autocatalysis and Autoinhibition Oscillatory Reactions, in: Putanov, P. New Chalenges in Catalysis, (Ed.), SANU, Novi Sad, 139-162.
  53. [53]  Benson, S.W. (1960), The Foundations of the Chemical Kinetics, McGraw Hill Book Company Inc., New York.
  54. [54]  Yeremin, E.N. (1979), The Foundations of the Chemical Kinetics, MIR, Moscow.
  55. [55]  Boissonade, J. and De Kepper, P. (1980), Transition from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system, Journal of Chemical Physics, 84, 501- 506.
  56. [56]  Andronov, A.A, Vitt, A.A., and Khaikin, S.E (1966), Theory of Oscillations, Pergamon, New York.
  57. [57]  Maselko, J.J. (1982), Determination of Bifurcation in Chemical Systems. An Experimental Method, Journal of Chemical Physics 67, 17-26.
  58. [58]  Noszticzius, Z., Stirling, P., and Wittman, M. (1985), Measurement of Bromine Removal Rate in the Oscillatory BZ Reaction of Oxalic Acid. Transition from Limit Cycle Oscillations to Excitability via Saddle-Node Infinite Period Bifurcation, Journal of Chemical Physics, 89, 4914-4942.
  59. [59]  Gáspár, V. and Galambosi, P. (1986), Bifurcation Diagram of the Osclllatory Belousov-Zhabotinskii System of Oxalic Acid in a Continuous Flow Stirred Tank Reactor. Further Possible Evidence of Saddle Node Infinite Period Bifurcation Behavior of the System, Journal of Chemical Physics, 90, 2222-2226.
  60. [60]  Noszticzius, Z., Wittman, M., and Stirling, P. (1987), Bifurcation from Excitability to Limit Cycle Oscillations at the End of the Induction Period in the Classical Belousov-Zhabotinsky Reaction, Journal of Chemical Physics, 86, 1922-1926.
  61. [61]  Bar-Eli, K. and Brøns, M. (1990), Period lengthening near the end of oscillations in chemical systems, Journal of Chemical Physics, 94, 7170-7177.
  62. [62]  Gaspard, P. (1990), Measurement of the Instability Rate of a Far-from-Equilibrium Steady State at an Infinite Period Bifurcation, Journal of Chemical Physics, 94, 1-3.
  63. [63]  Vukojević, V., Anić, S. and Kolar-Anić, Lj. (2000), Investigation of the dynamics behavior of the Bray- Liebhafsky reaction in the CSTR. Determination of the bifurcation points, Journal of Chemical Physics A , 104, 10731-10739.
  64. [64]  Pejić, N,. Maksimović, J., Ribič, D., and Kolar-Anić, Lj. (2009), Dynamic states of the Bray-Liebhafsky reaction when sulfuric acid is the control parameter, Russian Journal of Physical Chemistry , 83(9), 1666-1671.
  65. [65]  Pejić, N., Vujković, M, Maksimović, J., Ivanović, A., Anić, S., Čupić, Ž., and Kolar-Anić, Lj. (2011), Dynamic behavior of the Bray-Liebhafsky oscillatory reaction controlled by sulfuric acid and temperature, Russian Journal of Physical Chemistry, 85(13), 66-72.
  66. [66]  Noszticzius, Z. (1987), Bifurcation from excitability to limit cycle oscillations at the end of the induction period in the classical Belousov-Zhabotinsky reaction, Journal of Chemical Physics, 86, 1922-1926.
  67. [67]  Maselko, J. and Swinney, H.L. (1986), Complex periodic oscillations and Farey arithmetic in the Belousov- Zhabotinskii reaction, Journal of Chemical Physics, 85, 6430-6441.
  68. [68]  Petrov, V., Scott, S.K., and Showalter, K. (1992), Mixed-mode oscillations in chemical systems, Journal of Chemical Physics, 97, 6191-6198.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates