|
Spatial chaotic structure of attractors of reaction-diffusion systems
Author(s):
V.
Afraimovich;
A.
Babin;
S.-N.
Chow
Journal:
Trans. Amer. Math. Soc.
348
(1996),
5031-5063.
MSC (1991):
Primary 35K57;
Secondary 34C35
MathSciNet review:
1344202
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
The dynamics described by a system of reaction-diffusion equations with a nonlinear potential exhibits complicated spatial patterns. These patterns emerge from preservation of homotopy classes of solutions with bounded energies. Chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree. This group corresponds to level sets of the potential. The estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem.
References:
- [A]
- V. M. Alekseev, Symbolic Dynamics, XI Summer School of Academy of Science of Ukraine, Kiev, Naukova Dumka (1976). MR 57:4249
- [AC]
- V. S. Afraimovich and S.-N. Chow, Criteria of spatial chaos in lattice dynamical systems, Preprint CDSNS 93-142 (1993).
- [ACZ]
- A. Ambrosetti, V. Coti-Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Rat. Mech. Anal. 112 (1990), 339-362. MR 91k:34053
- [BBH]
- F. Bethuel, N. Brezis and F. Helein, Ginzburg-Landau Vortices, Birkhäuser, 1994. MR 95c:58044
- [BV]
- A. V. Babin, M. I. Vishik, Attractors of Evolution Equations, North-Holland, 1991. MR 92j:35016
- [B]
- A. Bahri, The variational contribution of the periodic orbits provided by Birkhoff-Lewis Theorem (with applications to convex Hamiltonians and to three-body type problems), Duke Math. J. 70 (1993), 1-205. MR 94g:58036
- [FFG]
- A. T. Fomenko, D. G. Fuchs, V. V. Gutenmacher, Homotopic Topology, Academiai Kiado, Budapest, 1986. MR 88f:55001
- [G]
- W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204 (1975), 113-135. MR 51:14152
- [Ha]
- J. K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys and Monographs, vol. 25, AMS, 1988. MR 89g:58059
- [Ham]
- R. Hamilton, Harmonic Maps of Manifolds with Boundary, Lecture Notes in Mathematics, vol. 471, Springer Verlag, 1975. MR 58:2872
- [H]
- D. Henry, Geometric Theory of Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, 1981. MR 83j:35084
- [JM]
- S. Jimbo and Y. Morita, Stability of nonconstant steady-state solutions to a Ginzburg-Landau equation in higher space dimensions, Nonlinear Anal. TMS 22 (1994), 753-770. CMP 94:10
- [KW]
- K. Kishimoto and M. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction diffusion systems on convex domain, J. Differential Equations 58 (1985), 15-21. MR 87a:35105
- [L]
- J. L. Lions, Quelques méthodes de résolution des problemes aux limites non lineaires, Dunod, Paris, 1969. MR 41:4326
- [M]
- W. S. Massey, Algebraic Topology: An Introduction, Harcourt Brace and World, Inc., New York, 1967. MR 35:2271
- [N]
- Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit, Dynamics Reported New Series, No. 3 (1994), 25-103.
- [R]
- P. M. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184. MR 57:1674
- [W]
- G. W. Whitehead, Elements of Homotopy Topology, Springer-Verlag, 1978. MR 80b:55001
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (1991):
35K57,
34C35
Retrieve articles in all Journals with
MSC (1991):
35K57,
34C35
Additional Information:
V.
Afraimovich
Affiliation:
CDSNS, Georgia Institute of Technology, Atlanta, Georgia 30332-0190
A.
Babin
Affiliation:
Moscow State University of Communications, Obraztsova 15, 101475 Moscow, Russia
S.-N.
Chow
Affiliation:
CDSNS and School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
DOI:
10.1090/S0002-9947-96-01578-4
PII:
S 0002-9947(96)01578-4
Keywords:
Reaction-diffusion system; potential; homotopy complexity; symbolic dynamics
Received by editor(s):
July 18, 1994
Received by editor(s) in revised form:
June 22, 1995
Additional Notes:
The first and third authors were partially supported by ARO DAAH04-93G-0199.
Research was partially supported by NIST Grant 60NANB2D1276.
Copyright of article:
Copyright
1996,
American Mathematical Society
|
