Diffusion Chaos in Reaction – Diffusion Boundary Problem in the Dumbbell Domain
https://doi.org/10.18255/1818-1015-2013-3-43-57
Abstract
We consider a boundary problem of reaction-diffusion type in the domain consisting of two rectangular areas connected by a bridge. The bridge width is a bifurcation parameter of the problem and is changed in such way that the measure of the domain is preserved. The conditions on chaotic oscillations emergence were studied and the dependence of invariant characteristics of the attractor on the bridge width was constructed. The diffusion parameter was chosen such that in the case of widest possible bridge (corresponding to a rectangular domain) the spatially homogeneous cycle of the problem is orbitally asymptotically stable. By decreasing the bridge width the homogeneous cycle looses stability and then the spatially inhomogeneous chaotic attractor emerges. For the obtained attractor we compute Lyapunov exponents and Lyapunov dimension and notice that the dimension grows as the parameter decreases but is bounded. We show that the dimension growth is connected with the growing complexity of stable solutions distribution with respect to the space variable.
About the Authors
S. D. Glyzin
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
д-р физ.-мат. наук, зав. кафедрой компьютерных сетей,
Sovetskaya str., 14, Yaroslavl, 150000, Russia
P. L. Shokin
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
научный сотрудник,
Sovetskaya str., 14, Yaroslavl, 150000, Russia
References
1. Turing Alan M. The Chemical Basis of Morphogenesis // Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences. V. 237. No. 641 (Aug. 14, 1952).
2. Nicolis G., Prigogine I. Self-Organization in Non-Equilibrium Systems. Wiley, 1977.
3. Kuramoto Y. Diffusion-Induced Chaos in Reaction Systems // Prog. Theor. Phys. Supplement. 1978. No. 64(1978). P. 346–367. DOI : 10.1143/PTPS.64.346.
4. Arrieta J. M. Neumann eigenvalue problems on exterior perturbations of the domain // Journal of Differential Equations. 1995. V. 118, No. 1. P. 54–103.
5. Arrieta J. M. Rates of Eigenvalues on a Dumbbell Domain. Simple Eigenvalue Case // Transactions of the American Mathematical Society. 1995. V. 347, No. 9 (Sep., 1995). P. 3503–3531.
6. Arrieta J. M., Carvalho A. N., Lozada-Cruz G. Dynamics in dumbbell domains I. Continuity of the set of equilibria // Journal of Differential Equations. 2006. V. 231, No. 2. P. 551–597.
7. Arrieta J. M., Carvalho A. N., Lozada-Cruz G. Dynamics in dumbbell domains II. The limiting problem // Journal of Differential Equations. 2009. V. 247, No 1. P. 174–202.
8. Arrieta J. M., Carvalho A. N., Lozada-Cruz G. Dynamics in dumbbell domains III. Continuity of attractors // Journal of Differential Equations. 2009. V. 247, No. 1. P. 225–259.
9. Васильева А. Б., Кащенко С. А., Колесов Ю. С., Розов Н. Х. Бифуркация автоколебаний нелинейных параболических уравнений с малой диффузией // Математический сборник. 1986. Т. 130(172), № 4(8). С. 488–499. (English transl.: Vasil’eva A. B., Kashchenko S.A., Kolesov Yu.S., Rozov N.Kh. Bifurcation of self-oscillations of nonlinear parabolic equations with small diffusion // Mathematics of the USSR-Sbornik, 1987. V 58:2. P. 491–503.)
10. Dormand J.R., Prince P.J. A Family of Embedded Runge – Kutta Formulae // J. Comp. Appl. Math. 1980. V. 6. P. 19–26.
11. Оселедец В.И. Мультипликативная эргодическая теорема. Характеристические показатели Ляпунова динамических систем // Тр. ММО. Т. 19. М., 1968. С. 179–210. (Oseledec V.I. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems // Trudy Moskov. Mat. Obs. V. 19. 1968. P. 179–210 [in Russian].)
12. Benettin G., Galgani L., Strelcyn J. M. Kolmogorov entropy and numerical experiments // Phys. Rev. 1976. V. A14. P. 2338–2345.
13. Wolf A., Swift J. B., Swinney H. L., Vastano J. A. Determining Lyapunov exponents from a time series // Physica D. 1985. V. D16. P. 285–317.
14. Глызин Д. С., Глызин С. Д., Колесов А.Ю., Розов Н. Х. Метод динамической перенормировки для нахождения максимального ляпуновского показателя хаотического аттрактора // Дифференциальные уравнения. 2005. Т. 41, № 2. С. 268–273.
15. (English transl.: Glyzin D.S., Glyzin S.D., Kolesov A.Yu., and Rozov N.Kh. The Dynamic Renormalization Method for Finding the Maximum Lyapunov Exponent of a Chaotic Attractor // Differential Equations. 2005. V. 41. No. 2. P. 284–289.)
16. Frederickson P., Kaplan J., Yorke J. The Lyapunov dimension of strange attractors // J. Different. Equat. 1983. V. 49. №2. P. 185–207.
17. Глызин С. Д. Разностные аппроксимации уравнения «реакция-диффузия» на отрезке // Моделирование и анализ информационных систем. 2009. Т. 16, № 3. С. 96–116. (Glyzin S. D. Difference approximations of “reaction – diffusion” equation on a segment // Modeling and Analysis of Information Systems. 2009. V. 16, No 3. P. 96–116 [in Russian].)
18. Глызин С.Д., Колесов А.Ю., Розов Н.Х. Конечномерные модели диффузионного хаоса // Журнал вычислительной математики и математической физики. 2010. Т. 50, № 5. С. 860–875. (English transl.: Glyzin S.D., Kolesov A.Yu., and Rozov N.Kh. Finitedimensional models of diffusion chaos // Computational Mathematics and Mathematical Physics. 2010. V. 50. No 5. P. 816–830. DOI: 10.1134/S0965542510050076.)
19. Глызин С. Д. Размерностные характеристики диффузионного хаоса // Моделирование и анализ информационных систем. 2013. Т. 20, № 1. С. 30–51. (Glyzin S. D. Dimensional Characteristics of Diffusion Chaos // Modeling and Analysis of Information Systems. 2013. V. 20, No 1. P. 30–51 [in Russian].)
Review
For citations:
Glyzin S.D., Shokin P.L. Diffusion Chaos in Reaction – Diffusion Boundary Problem in the Dumbbell Domain. Modeling and Analysis of Information Systems. 2013;20(3):43-57. (In Russ.) https://doi.org/10.18255/1818-1015-2013-3-43-57
Views:
1018
Email this article 