Morse decomposition, attractors and chain recurrence
DOI:
https://doi.org/10.4067/S0716-09172006000100006Keywords:
Morse decomposition, attractors, repellers, chains, invariants, descomposición Morse, atractores, repulsores, cadenas, invariantes.Abstract
The global behavior of a dynamical system can be described by its Morse decompositions or its attractor and repeller configurations. There is a close relation between these two approaches and also with (maximal) chain recurrent sets that describe the system behavior on finest Morse sets. These sets depend upper semicontinuously on parameters. The connection with ergodic theory is provided through the construction of invariant measures based on chains.References
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[2] J.-P. Aubin and H. Frankowska, Set Valued Analysis, Birkhäuser, (1990).
[3] J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99, pp. 332-334, (1992).
[4] C. Carathéodory, Vorlesungen über reelle Funktionen, Leipzig, 1918. Reprint: Chelsea Publ. Co., 1968.
[5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LN Mathematics vol. 580, Springer-Verlag, (1977).
[6] F. Colonius, R. Fabbri, and R. Johnson, Chain recurrence, growth rates and ergodic limits, Ergodic Theory and Dynamical Systems, to appear.
[7] F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, (2000).
[8] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series no. 38, Amer. Math. Soc., (1978).
[9] C. Conley, The gradient structure of a flow I, Ergodic Theory Dyn. Syst. 8 (1988), 11-26. Original: IBM Research, RC 3939, Yorktown Heights, N. Y., 1972.
[10] R. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin/Cummings, (1986).
[11] R. Easton, Geometric Methods for Discrete Dynamical Systems, Oxford University Press, (1998).
[12] F. Engelking, General Topology, PWN-Polish Scientific, Warszawa, (1977).
[13] E. Hewitt and K. Stromberg, Real and Abstract Analysis, SpringerVerlag, (1965).
[14] M. Hurley, Chain recurrence, semiflows, and gradients, J. Dyn. Diff. Equations 7, pp. 437-456, (1995).
[15] A. Katok and B. Hasselblatt, An Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, (1995).
[16] R. Mañé, Ergodic Theory and Differentiable Dynamics, SpringerVerlag, (1987).
[17] V.I. Natanson, Theorie der Funktionen einer reellen Veränderlichen, Second Edition, Akademie-Verlag, Berlin, 1961. Russian original 1957.
[18] V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Dynamical Systems, Princeton University Press, 1960. Russian original 1949.
[19] G.K. Pedersen, Analysis Now, Springer-Verlag, (1988).
[20] S. Pilyugin, The Space of Dynamical Systems with the C°-Topology, Springer-Verlag, (1994).
[21] M. Pollicott, Lectures on Ergodic Theory and Pesin Theory of Compact Manifolds, London Mathematical Society Lect. Notes Series 180, Cambridge University Press, (1993).
[22] C. Robinson, Dynamical Systems, 2nd. Edition, CRC Press Inc., (1998).
[23] H.L. Royden, Real Analysis, Second Edition, Macmillan Publishing Co., (1968).
[24] K.P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, (1987).
[25] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291, pp. 1-41, (1985).
[26] D. Salamon and E. Zehnder, Flows on vector bundles and hyperbolic sets, Trans. Amer. Math. Soc. 306, pp. 623-649, (1988).
[27] N. Scherbina, Continuity of one-parameter families of sets, Dokl. Ak. Nauk SSSR 234, pp. 327-329. In Russian, (1977).
[28] D. Szolnoki, Set oriented methods for computing reachable sets and control sets, Discrete and Continuous Dynamical Systems Series B 3, pp. 361-382, (2003).
[29] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, (1972).
Published
2017-05-08
How to Cite
[1]
J. Ayala-Hoffmann, P. Corbin, K. McConville, F. Colonius, W. Kliemann, and J. R. Peters, “Morse decomposition, attractors and chain recurrence”, Proyecciones (Antofagasta, On line), vol. 25, no. 1, pp. 79-109, May 2017.
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