ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Karavaev A. D., Ryzhkov A. B., Kazakov V. P. Birth and death of fractal tore in the Belousov - Zhabotinsky reaction model. Izvestiya VUZ. Applied Nonlinear Dynamics, 2001, vol. 9, iss. 1, pp. 89-100. DOI: 10.18500/0869-6632-2001-9-1-89-100

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2001no3p089.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
541.127:541.515:519.6
DOI: 
10.18500/0869-6632-2001-9-1-89-100

Birth and death of fractal tore in the Belousov - Zhabotinsky reaction model

Autors: 
Karavaev Alexandr Dmitrievich, Institute of Organic Chemistry of the UNTs RAS
Ryzhkov Andrei Borisovich, Institute of Organic Chemistry of the UNTs RAS
Kazakov Valerii Petrovich, Institute of Organic Chemistry of the UNTs RAS
Abstract: 

The mechanism of birth and destruction of chaotic toroidal attractor — fractal tore — is investigated for the 11—stage Belousov — Zhabotinsky reaction model. It is revealed, that fractal tore emerges as а result of period—doubling bifurcations cascade of а resonant state оп torus, and disappears through type I intermittency. Constructed bifurcation diagram shows, that fractal toris exist in a wide enough range, where resonant states, fractal toris and areas of intermittency appear conformingly in turn. It gives the basis to believe, that observed model dynamics, as the Belousov — Zhabotinsky reaction itself, involves two fundamental frequencies, and that the evolution of described regimes occurs on torus upon general tendency of rotation number to reduction.

Key words: 
-
Acknowledgments: 
The authors are grateful to G.G. Malinetsky for providing the KORDIM program, E.A. Novikov for the (m,k)-method algorithm, and A.B. Potapov for help in calculating the correlation dimension and Lyapunov indices.
Reference: 
  1. Field R, Burger M, editors. Fluctuations and traveling waves in chemical systems.Moskow: Mir, 1988. 720 p.
  2. Argoul F, Arneodo А, Richetti P. Roux JC, Swinney HL. Chemical chaos: from hints 10 confirmation. Acc. Chem. Res. 1987;20(12):436–442.
  3. Schuster G. Ceterministic Chaos. Moscow: Mir, 1988. 240 p.
  4. Vidal C, Pacault A, editors. Nonlinear Phenomena in Chemical Dynamics. Berlin: Springer-Verlag; 1981.
  5. Argoul F, Arneodo А, Richertti Р, Roux JC. From quasiperiodicity to chaos in the Belousov-Zhabotinskii reaction. I. Experiment. J. Chem. Phys. 1987;86(6):3325. DOI: 10.1063/1.452751.
  6. Argoul F, Arneodo А, Richetti P. J. Dynamique symbolique dans lа reaction de Belousov-Zhabotinskii: une illustration experimentale le lа theorie de Shil’nikov des orbites homoclines. J. Chim. Phys. 1987;84(11-12):1367.
  7. Richetti P, Roux JC, Argoul F, Arneodo A. From quasiperiodicity to chaos in the Belouso-Zhabotinskii reaction. II. Modeling and theory. J. Chem. Phys. 1987;86(6):3339–3356. DOI: 10.1063/1.451992.
  8. Noskov OB, Karavaev AD, Spivak CH, Kazakov VP. Modeling of complex dynamics of the Belousov-Zhabotinsky reaction: the crucial role of fast variables. Kinetics and catalysis. 1992;33(3):704–712.
  9. Noskov OV, Karavaev AD, Kazakov VP, Spivak SI. Chaos in simulated Belousov-Zhabotinsky Reaction. Mend. Commun. 1994;4:82. DOI: 10.1038/s41598-020-77874-6.
  10. Noskov OV, Karavaev AD, Kazakov VP, Spivak SI. Quasiperiodic to bursting oscillations transition in thе model оf the Belousov-Zhabotinsky reaction. Mend. Commun. 1997;1:27.
  11. Karavaev AD, Ryzhkov AB, Noskov OB, Kazakov VP. Observation of a fractal torus in the Belousov-Zhabotinsky reaction model. Moskow: DAN USSR. 1998;363(1):71.
  12. Aronson DG, Chory MA, Най GR, МсСейее RP. Bifurcations from аn invariant circle for two-parameter families of maps of the plane: computer-assisted study. Commun. Math. Phys. 1982;83(3):303. DOI: 10.1007/BF01213607.
  13. Ruoff P, Noyes RM. An amplified oregonator model simulating alternative excitabilities, transitions in types of oscillations, and temporary bistability in a closed system. J. Chem. Phys. 1986;84(3):1413–1423. DOI: 10.1063/1.450484.
  14. Novikov EA, Golushko ML, Shitiov YA. Approximation оf Jacobi matrix in the (m,k) - method оf order three. Advanses in Modeling & Analysis. USA: AMSE Press. 1995;28(3):19.
  15. Malinetsky GG, Potapov AB. On Calculation of Dimensions of Strange Attractors. Moscow: Prepr. IPM AS. 1987.
  16. Malinetsky GG, Potapov AB. About calculation of dimensions of strange attractors. Zhurnal. comput. matem. i matem. phys. 1988;28(7):1021–1037.
  17. Wolf А, Swift JB, Swinney HL, Vastano JА. Determining Lyapunov exponents from а time series. Phisica D. 1985;16(3):285–317. DOI:10.1016/0167-2789(85)90011-9.
  18. Ryzhkov AB, Noskov OB, Karavaev AD, Kazakov VP. Stationaries and bifurcations of the Belousov-Zhabotinsky reaction. Mat. modeling. 1998;10(2):71.
  19. Anishchenko BC, Vadivasova TE, Astakhov VV. Nonlinear dynamics of chaotic and stochastic systems. Saratov: Izd. of Saratov University, 1999.
  20. Noskov OB, Karavaev AD, Kazakov VN. Homoclinics in the model of the Belousov-Zhabotinsky reaction. Moskow: DAN USSR. 1997;353(6):774-777.
Received: 
13.09.2000
Accepted: 
14.11.2000
Published: 
05.06.2001
  • 152 reads