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Discretization strategies for computing Conley indices and Morse decompositions of flows
Towards a formal tie between combinatorial and classical vector field dynamics
1. | Département de mathématiques, Université de Sherbrooke, 2500 boul. Université, Sherbrooke, Qc, J1K2R1, Canada |
2. | Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. St. Łojasiewicza 6, 30-348 Kraków, Poland |
3. | Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030 |
References:
[1] |
M. Allili and T. Kaczynski, An algorithmic approach to the construction of homomorphisms induced by maps in homology, Transactions of the American Mathematical Society, 352 (2000), 2261-2281.
doi: 10.1090/S0002-9947-99-02527-1. |
[2] |
B. Batko and M. Mrozek, Weak index pairs and the Conley index for discrete multivalued dynamical systems, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1143-1162.
doi: 10.1137/15M1046691. |
[3] |
C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Society, Providence, RI, 1978. |
[4] |
C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Transactions of the American Mathematical Society, 158 (1971), 35-61.
doi: 10.1090/S0002-9947-1971-0279830-1. |
[5] |
R. Forman, Morse theory for cell complexes, Advances in Mathematics, 134 (1998), 90-145.
doi: 10.1006/aima.1997.1650. |
[6] |
R. Forman, Combinatorial vector fields and dynamical systems, Mathematische Zeitschrift, 228 (1998), 629-681.
doi: 10.1007/PL00004638. |
[7] |
L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, $2^{nd}$ ed, Topological Fixed Point Theory and Its Applications, 4, Springer Verlag, The Netherlands, 2006. |
[8] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer Verlag, New York, NY, 2004.
doi: 10.1007/b97315. |
[9] |
T. Kaczynski and M. Mrozek, Conley index for discrete multivalued dynamical systems, Topology and Its Applications, 65 (1995), 83-96.
doi: 10.1016/0166-8641(94)00088-K. |
[10] |
H. King, K. Knudson and N. Mramor, Generating discrete Morse functions from point data, Experimental Mathematics, 14 (2005), 435-444.
doi: 10.1080/10586458.2005.10128941. |
[11] |
M. Mrozek and B. Batko, Coreduction homology algorithm, Discrete and Computational Geometry, 41 (2009), 96-118.
doi: 10.1007/s00454-008-9073-y. |
[12] |
V. Robins, P. J. Wood and A. P. Sheppard, Theory and algorithms for constructing discrete Morse complexes from grayscale digital images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 1646-1658.
doi: 10.1109/TPAMI.2011.95. |
[13] |
T. Stephens and T. Wanner, Rigorous validation of isolating blocks for flows and their Conley indices, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1847-1878.
doi: 10.1137/140971075. |
show all references
References:
[1] |
M. Allili and T. Kaczynski, An algorithmic approach to the construction of homomorphisms induced by maps in homology, Transactions of the American Mathematical Society, 352 (2000), 2261-2281.
doi: 10.1090/S0002-9947-99-02527-1. |
[2] |
B. Batko and M. Mrozek, Weak index pairs and the Conley index for discrete multivalued dynamical systems, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1143-1162.
doi: 10.1137/15M1046691. |
[3] |
C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Society, Providence, RI, 1978. |
[4] |
C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Transactions of the American Mathematical Society, 158 (1971), 35-61.
doi: 10.1090/S0002-9947-1971-0279830-1. |
[5] |
R. Forman, Morse theory for cell complexes, Advances in Mathematics, 134 (1998), 90-145.
doi: 10.1006/aima.1997.1650. |
[6] |
R. Forman, Combinatorial vector fields and dynamical systems, Mathematische Zeitschrift, 228 (1998), 629-681.
doi: 10.1007/PL00004638. |
[7] |
L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, $2^{nd}$ ed, Topological Fixed Point Theory and Its Applications, 4, Springer Verlag, The Netherlands, 2006. |
[8] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer Verlag, New York, NY, 2004.
doi: 10.1007/b97315. |
[9] |
T. Kaczynski and M. Mrozek, Conley index for discrete multivalued dynamical systems, Topology and Its Applications, 65 (1995), 83-96.
doi: 10.1016/0166-8641(94)00088-K. |
[10] |
H. King, K. Knudson and N. Mramor, Generating discrete Morse functions from point data, Experimental Mathematics, 14 (2005), 435-444.
doi: 10.1080/10586458.2005.10128941. |
[11] |
M. Mrozek and B. Batko, Coreduction homology algorithm, Discrete and Computational Geometry, 41 (2009), 96-118.
doi: 10.1007/s00454-008-9073-y. |
[12] |
V. Robins, P. J. Wood and A. P. Sheppard, Theory and algorithms for constructing discrete Morse complexes from grayscale digital images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 1646-1658.
doi: 10.1109/TPAMI.2011.95. |
[13] |
T. Stephens and T. Wanner, Rigorous validation of isolating blocks for flows and their Conley indices, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1847-1878.
doi: 10.1137/140971075. |
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