Given an unknown attractor 𝓐 in a continuous dynamical system, how can we discover the topology and dynamics of 𝓐? As a practical matter, how can we do so from only a finite amount of information? One way of doing so is to produce a semi-conjugacy from 𝓐 onto a model system 𝓜 whose topology and dynamics are known. The complexity of 𝓜 then provides a lower bound for the complexity of 𝓐. The Conley index can be used to construct a simplicial model and a surjective semi-conjugacy for a large class of attractors. The essential features of this construction are that the model 𝓜 can be explicitly described; and that the finite amount of information needed to construct it is computable.
@article{bwmeta1.element.bwnjournal-article-bcpv47i1p145bwm,
author = {McCord, Christopher},
title = {Reconstructing the global dynamics of attractors via the Conley index},
journal = {Banach Center Publications},
volume = {50},
year = {1999},
pages = {145-156},
zbl = {0942.37003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv47i1p145bwm}
}
McCord, Christopher. Reconstructing the global dynamics of attractors via the Conley index. Banach Center Publications, Tome 50 (1999) pp. 145-156. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv47i1p145bwm/
[000] [1] M. Carbinatto, J. Kwapisz and K. Mischaikow, Horseshoes and the Conley index spectrum, preprint. | Zbl 0971.37005
[001] [2] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Lecture Notes 38 A.M.S. Providence, R.I., 1978.
[002] [3] A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Erg. Th. & Dyn. Sys. 7 (1987), 93-103. | Zbl 0633.58040
[003] [4] J. Folkman, The homology group of a lattice, J. Math. & Mech. 15 (1966), 631-636. | Zbl 0146.01602
[004] [5] T. Gedeon, Cyclic feedback systems, to appear in Mem. Amer. Math. Soc.
[005] [6] T. Gedeon and K. Mischaikow, Structure of the global attractor of cyclic feedback systems, J. Dynamics and Diff. Eq. 7 (1995), 141-190. | Zbl 0823.34057
[006] [7] T. Kaczynski and M. Mrozek, Conley index for discrete multi-valued dynamical systems, Topology Appl. 65 (1995), 83-96. | Zbl 0843.54042
[007] [8] C. McCord, Simplicial models for the global dynamics of attractors, preprint.
[008] [9] C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, Jour. Amer. Math. Soc. 9 (1996), 1095-1133. | Zbl 0861.58023
[009] [10] K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations, SIAM J. Math. Anal. 26 (1995), 1199-1224. | Zbl 0841.35014
[010] [11] K. Mischaikow and Y. Morita, Dynamics on the global attractor of a gradient flow arising from the Ginzburg-Landau equation, Japan J. Indust. Appl. Math. 11 (1994), 185-202. | Zbl 0807.58029
[011] [12] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), 205-236. | Zbl 0840.58033
[012] [13] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computer-assisted proof, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 66-72. | Zbl 0820.58042
[013] [14] K. Mischaikow, M. Mrozek, J. Reiss and A. Szymczak, From time series to symbolic dynamics: An algebraic topological approach, preprint.
[014] [15] M. Mrozek, From the theorem of Ważewski to computer assisted proofs in dynamics, Panoramas of mathematics (Warsaw, 1992/1994), 105-120, Banach Center Publ. 34, Polish Acad. Sci., Warsaw, 1995. | Zbl 0848.34032
[015] [16] M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs in dynamics, Comput. Math. Appl. 32 (1996), 83-104. | Zbl 0861.58027
[016] [17] A. Szymczak, A combinatorial procedure for finding isolating neighborhoods and index pairs, Proc. Royal Soc. Edinburgh 127A (1997), 1075-1088. | Zbl 0882.54032