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

For citation:

Ryashko L. B., Smirnov A. V. Deterministic and stochastic stability analysis for glycolitic oscillator. Izvestiya VUZ. Applied Nonlinear Dynamics, 2005, vol. 13, iss. 6, pp. 99-112. DOI: 10.18500/0869-6632-2005-13-5-99-112

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 2005no5-6p099.pdf
(downloads: 199)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
531.36
DOI: 
10.18500/0869-6632-2005-13-5-99-112

Deterministic and stochastic stability analysis for glycolitic oscillator

Autors: 
Ryashko Lev Borisovich, Ural Federal University named after the first President of Russia B.N.Yeltsin
Smirnov Aleksej Valerevich, Ural Federal University named after the first President of Russia B.N.Yeltsin
Abstract: 

The methods of sensitivity analysis of cycles under deterministic and stochastic disturbances for Higgins model describing glycolytic self-oscillations are considered. Two approaches connected with local exponents and stochastic sensitivity function are compared. The most sensitive parts of cycles are discovered. It was found that some parts of cycle lose stochastic stability along with stability increasing of cycle as whole.

Key words: 
-
Reference: 
  1. Higgins J. The theory of oscillating reactions - kinetics symposium. J. Ind. Eng. Chem. 1967;59(5):18–62. DOI: 10.1021/ie50689a006.
  2. Higgins J. A chemical mechanism for oscillation of glycolytic intermediates in yeast cells. Proc. Natl. Acad. Sci. USA. 1964;51(6):989–994. DOI: 10.1073/pnas.51.6.989.
  3. Sel’kov EE. Self-oscillations in glycolysis 1. A simple kinetic model. Eur. J. Biochem. 1968;4(1):79–86. DOI: 10.1111/j.1432-1033.1968.tb00175.x.
  4. Romanovsky YM, Stepanova NV, Chernavsky DS. Mathematical Modeling in Biophysics. Moscow: Nauka; 1975. 344 p. (in Russian).
  5. Gurel G, Gurel O. Oscillations in Chemical Reactions. Berlin: Springer; 1986. 124 p. DOI: 10.1007/3-540-12575-2.
  6. Tomita К, Daido H. Possibility of chaotic behaviour and multi-basins in forced glycolytic oscillations. Physics Letters A. 1980;79(2–3):133–137. DOI: 10.1016/0375-9601(80)90226-1.
  7. Kurrer C, Schulten K. Effect of noise and perturbations on limit cycle systems. Physica D. 1991;50(3):311–320. DOI: 10.1016/0167-2789(91)90001-P.
  8. Ali F, Menzinger M. On the local stability of limit cycle. Chaos. 1999;9(2):348–356. DOI: 10.1063/1.166412.
  9. Rytov SM. Fluctuations in Thomson-type self-oscillating systems. I. Sov. Phys. JETP. 1955;29(3):304–314 (in Russian).
  10. Rytov SM. Fluctuations in Thomson-type self-oscillating systems. II. Sov. Phys. JETP. 1955;29(3):315 (in Russian).
  11. Dyckman M, Chu X, Ross J. Stationary probability distribution near stable limit cycles far from Hopf bifurcation points. Physical Review E. 1993;48(3):1646–1654. DOI: 10.1103/physreve.48.1646.
  12. Kuznetsov AP, Kapustina YV. Scaling properties during the transition to chaos in model mappings with noise. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(6):78–87 (in Russian).
  13. Kopeikin AS, Vadivasova TE, Anischenko VS. Peculiarities of the process of establishing a probability measure on chaotic attractors in the Lorentz and Ressler systems taking into account fluctuations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(6):65–77 (in Russian).
  14. Freidlin MI, Wentzell AD. Random Perturbations of Dynamical Systems. New York: Springer; 1984. 328 p. DOI: 10.1007/978-1-4684-0176-9.
  15. Mil'Shtein GN, Ryashko LB. A first approximation of the quasipotential in problems of the stability of systems with random non-degenerate perturbations. Journal of Applied Mathematics and Mechanics. 1995;59(1):47–56. DOI: 10.1016/0021-8928(95)00006-B.
  16. Bashkirtseva IA, Ryashko LB. Sensitivity analysis of the stochastically forced Lorenz model cycles under period-doubling bifurcations. Dynamic Systems and Applications. 2002;11(2):293–309.
  17. Bashkirtseva IA, Ryashko LB. Sensitivity analysis of the stochastically and periodically forced Brusselator. Physica A. 2000;278(1–2):126–139. DOI: 10.1016/S0378-4371(99)00453-7.
  18. Bashkirtseva IA, Smirnov AV, Ryashko LB. Stochastic stability of the glycolytic oscillator. In: Proceedings of the III Ural Scientific and Practical Conference «Mathematical Modeling in Medicine and Biology»; 2001. P. 18 (in Russian).
  19. Bashkirtseva IA, Smirnov AV, Ryashko LB. Comparative analysis of the stability of the glycolytic oscillator to deterministic and random disturbances. In: Materials of the 33rd Regional Youth School-Conference «Problems of Theoretical and Applied Mathematics»; 2002. P. 107 (in Russian).
Received: 
01.11.2003
Accepted: 
03.11.2004
Published: 
28.02.2006
Short text (in English):
PDF icon a2005no5-6p099.doc_.pdf
(downloads: 88)
  • 1547 reads