Reconstructing the global dynamics of attractors via the Conley index

• Autorzy: Christopher McCord
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Conley index; semi-conjugacy; simplicial complex; attractor

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1999
• Tom: 47
• Numer: 1
• Strony: 145-156

Daty

wydano

1999

Twórcy

autor

Christopher McCord

• Institute for Dynamics, Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025, U.S.A.

Bibliografia

• [1] M. Carbinatto, J. Kwapisz and K. Mischaikow, Horseshoes and the Conley index spectrum, preprint.
• [2] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Lecture Notes 38 A.M.S. Providence, R.I., 1978.
• [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.
• [4] J. Folkman, The homology group of a lattice, J. Math. & Mech. 15 (1966), 631-636.
• [5] T. Gedeon, Cyclic feedback systems, to appear in Mem. Amer. Math. Soc.
• [6] T. Gedeon and K. Mischaikow, Structure of the global attractor of cyclic feedback systems, J. Dynamics and Diff. Eq. 7 (1995), 141-190.
• [7] T. Kaczynski and M. Mrozek, Conley index for discrete multi-valued dynamical systems, Topology Appl. 65 (1995), 83-96.
• [8] C. McCord, Simplicial models for the global dynamics of attractors, preprint.
• [9] C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, Jour. Amer. Math. Soc. 9 (1996), 1095-1133.
• [10] K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations, SIAM J. Math. Anal. 26 (1995), 1199-1224.
• [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.
• [12] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), 205-236.
• [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.
• [14] K. Mischaikow, M. Mrozek, J. Reiss and A. Szymczak, From time series to symbolic dynamics: An algebraic topological approach, preprint.
• [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.
• [16] M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs in dynamics, Comput. Math. Appl. 32 (1996), 83-104.
• [17] A. Szymczak, A combinatorial procedure for finding isolating neighborhoods and index pairs, Proc. Royal Soc. Edinburgh 127A (1997), 1075-1088.