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1999 | 47 | 1 | 9-19
Tytuł artykułu

The Conley index theory: A brief introduction

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A brief introduction to the Conley index theory is presented. The emphasis is the fundamental ideas of Conley's approach to dynamical systems and how it avoids some of the difficulties inherent in the study of nonlinear systems.
Słowa kluczowe
Rocznik
Tom
47
Numer
1
Strony
9-19
Opis fizyczny
Daty
wydano
1999
Twórcy
  • Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A.
Bibliografia
  • [1] M. Carbinatto, J. Kwapisz, and K. Mischaikow, Horseshoes and the Conley Index Spectrum, http://www.math.gatech.edu/ mischaik.
  • [2] M. Carbinatto and K. Mischaikow, Horseshoes and the Conley Index Spectrum (II): The theorem is sharp, http://www.math.gatech.edu/ mischaik.
  • [3] L. Arnold, C. Jones, K. Mischaikow, and G. Raugel, Dynamical Systems: Montecatini Terme, 1994, R. Johnson, ed. Lect. Notes in Math. 1609, Springer 1995.
  • [4] C. Conley, Isolated Invariant Sets and the Morse Index. CBMS Lecture Notes 38, A.M.S. Providence, R.I. 1978.
  • [5] C. Conley, A qualitative singular perturbation theorem, in: Global Theory of Dynamical Systems, (eds. Z. Nitecki and C. Robinson), Lecture Notes in Math. 819, Springer, 1980, 65-89.
  • [6] C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. AMS 158 (1971), 35-61.
  • [7] C. Conley and J. Smoller, Viscosity matrices for two dimensional nonlinear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 867-884.
  • [8] C. Conley and J. Smoller, Viscosity matrices for two dimensional nonlinear hyperbolic systems, II, Amer. J. Math. 94 (1972), 631-650.
  • [9] C. Conley and J. Smoller, On the structure of magnetohydrodynamic shock waves, Comm. Pure Appl. Math. 27 (1974), 367-375.
  • [10] C. Conley and J. Smoller, On the structure of magnetohydrodynamic shock waves, II, J. Math. Pures et Appl. 54 (1975), 429-444.
  • [11] M. Degiovanni and M. Mrozek, The Conley index for maps in the absence of compactness, Proc. Roy. Soc. Edin. 123A (1993), 75-94.
  • [12] A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Erg. Thy. Dyn. Sys. 7 (1987), 93-103.
  • [13] A. Floer, Proof of the Arnold conjecture for surfaces and generalizations to certain Kähler manifolds, Duke Math. J. 53 (1986), 1-32.
  • [14] J. Franks and D. Richeson, Shift Equivalence and the Conley Index, preprint 1998.
  • [15] R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. AMS 298 (1986), 193-213.
  • [16] R. Franzosa, The connection matrix theory for Morse decompositions, Trans. AMS 311 (1989), 561-592.
  • [17] R. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. AMS 310, (1988), 781-803.
  • [18] R. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, Journal of Differential Equations 71 (1988), 270-287.
  • [19] R. Franzosa and K. Mischaikow, Algebraic Transition Matrices in the Conley Index Theory, Trans. AMS 350 (1998), 889-912.
  • [20] T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, and J. Reineck, The Conley index for fast slow systems I: One dimensional slow variable, J. Dyn. Diff. Eq. (to appear).
  • [21] T. Kaczynski and M. Mrozek, Conley index for discrete multi-valued dynamical systems, Top. Appl. 65 (1995), 83-96.
  • [22] T. Kaczynski and M. Mrozek, Stable Index Pairs for Discrete Dynamical Systems, Bul. Can. Math. Soc. (to appear).
  • [23] C. McCord, Mappings and homological properties in the homology Conley index. Erg. Th. Dyn. Sys. 8* (1988), 175-198.
  • [24] C. McCord and K. Mischaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. AMS 333 (1992), 379-422.
  • [25] C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, Jour. AMS 9 (1996), 1095-1133.
  • [26] C. McCord, K. Mischaikow, and M. Mrozek. Zeta functions, periodic trajectories, and the Conley index, JDE 121 (1995), 258-292.
  • [27] K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations, SIAM Math. Anal. 26 (1995), 1199-1224.
  • [28] K. Mischaikow and Y. Morita, Dynamics on the Global Attractor of a Gradient Flow arising in the Ginzburg-Landau Equation, Jap. J. Ind. Appl. Math. 11 (1994), 185-202.
  • [29] K. Mischaikow and M. Mrozek, Isolating Neighborhoods and Chaos, Jap. J. Ind. Appl. Math. 12 (1995), 205-236.
  • [30] K. Mischaikow and M. Mrozek, Chaos in the Lorenz Equations: a Computer Assisted Proof, Bull. AMS 32 (1995), 66-72.
  • [31] K. Mischaikow and M. Mrozek, Chaos in the Lorenz Equations: a Computer Assisted Proof. Part II, Details, Math. Comp. 67 (1998), 1023-1046.
  • [32] K. Mischaikow, M. Mrozek, and J. Reineck, Singular Index Pairs, J. Dyn. Diff. Eq. (to appear).
  • [33] K. Mischaikow, M. Mrozek, J. Reiss, and A. Szymczak, Construction of Symbolic Dynamics from Experimental Time Series, http://www.math.gatech.edu/ mischaik.
  • [34] K. Mischaikow, M. Mrozek, A. Szymczak, and J. Reiss, From Time Series to Symbolic Dynamics: An Algebraic Topological Approach, http://www.math.gatech.edu/ mischaik.
  • [35] M. Mrozek, Leray functor and cohomological index for discrete dynamical systems, Trans. A. M. S. 318 (1990), 149-178.
  • [36] M. Mrozek, Topological invariants, multivalued maps, and computer assisted proofs, Comp. Math. 32 (1996), 83-104.
  • [37] M. Mrozek and K. P. Rybakowski, A cohomological Conley index for discrete dynamical systems, J. D. E. 90 (1991), 143-171.
  • [38] J. Reineck, Connecting orbits in one-parameter families of flows, Ergodic Theory and Dynamical Systems 8* (1988), 359-374.
  • [39] D. Richeson, Connection Matrix Pairs for the Discrete Conley Index, preprint 1997.
  • [40] J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory and Dynamical Systems 8* (1988), 375-393.
  • [41] J. W. Robbin and D. Salamon, Lyapunov maps, simplicial complexes and the Stone functor, Ergod. Th. Dyn. Sys. 12 (1992), 153-183.
  • [42] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag 1987.
  • [43] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. A.M.S. 291 (1985), 1-41.
  • [44] A. Szymczak, The Conley Index for Discrete Semidynamical Systems, Top. App. 66 (1995), 215-240.
  • [45] A. Szymczak, The Conley Index and Symbolic Dynamics, Topology 35 (1996), 287-299.
  • [46] A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. Roy. Soc. Edin., to appear.
  • [47] T. Ważewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Pol. Mat. 29 (1947), 279-313.
Typ dokumentu
Bibliografia
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