Abstract
Asymptotic and numerical analysis of relaxation self-oscillations in a three-dimensional system of Volterra ordinary differential equations that models the well-known Belousov reaction is carried out. A numerical study of the corresponding distributed model-the parabolic system obtained from the original system of ordinary differential equations with the diffusive terms taken into account subject to the zero Neumann boundary conditions at the endpoints of a finite interval is attempted. It is shown that, when the diffusion coefficients are proportionally decreased while the other parameters remain intact, the distributed model exhibits the diffusion chaos phenomenon; that is, chaotic attractors of arbitrarily high dimension emerge.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Finite-Dimensional Models of Diffusion Chaos,” Comput. Math. Math. Phys. 50, 816–830 (2010).
B. P. Belousov, “A Periodic Reaction and Its Mechanism,” in Autowave Processes in Systems with Diffusion (Institut prikl. fiz. AN SSSR, Gorky, 1981), pp. 176–186 [in Russian].
A. M. Zhabotinskii, Concentration Self-Oscillations (Nauka, Moscow, 1974) [in Russian].
Yu. S. Kolesov, “Adequacy Problem for Ecological Equations,” Available from VINITI, 1985, Yaroslavl’, no. 1901-85 [in Russian].
P. Ruoff and R. M. Noyes, “An Amplified Oregonator Model Simulating Alternative Excitabilities, Transitions in Types of Oscillations, and Temporary Bistability in a Closed System,” J. Chem. Phys. 84, 1413–1423 (1986).
J. Milnor, “On the Concept of Attractor,” Commun. Math. Phys. 99(2), 177–196 (1985).
P. Frederickson, J. Kaplan, and J. Yorke, “The Lyapunov Dimension of Strange Attractors,” J. Differ. Equations 49(2), 185–207 (1983).
Yu. S. Kolesov, “A Nonclassical Relaxation Cycle in a Three-Dimensional System of Differential Lotka-Volterra Equations,” Mat. Sb. 191(4), 91–106 (2000).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.D. Glyzin, A.Yu. Kolesov, N.Kh. Rozov, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 8, pp. 1400–1418.
Rights and permissions
About this article
Cite this article
Glyzin, S.D., Kolesov, A.Y. & Rozov, N.K. Relaxation oscillations and diffusion chaos in the Belousov reaction. Comput. Math. and Math. Phys. 51, 1307–1324 (2011). https://doi.org/10.1134/S0965542511080100
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542511080100