Towards a formal tie between combinatorial and classical vector field dynamics

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January 2016, 3(1): 17-50. doi: 10.3934/jcd.2016002

Towards a formal tie between combinatorial and classical vector field dynamics

Tomasz Kaczynski ^1, , Marian Mrozek ^2, and Thomas Wanner ^3,

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

Received September 2015 Revised June 2016 Published September 2016

Abstract
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Forman's combinatorial vector fields on simplicial complexes are a discrete analogue of classical flows generated by dynamical systems. Over the last decade, many notions from dynamical systems theory have found analogues in this combinatorial setting, such as for example discrete gradient flows and Forman's discrete Morse theory. So far, however, there is no formal tie between the two theories, and it is not immediately clear what the precise relation between the combinatorial and the classical setting is. The goal of the present paper is to establish such a formal tie on the level of the induced dynamics. Following Forman's paper [6], we work with possibly non-gradient combinatorial vector fields on finite simplicial complexes, and construct a flow-like upper semi-continuous acyclic-valued mapping on the underlying topological space whose dynamics is equivalent to the dynamics of Forman's combinatorial vector field on the level of isolated invariant sets and isolating blocks.

Keywords: multivalued maps, simplicial complex, discrete dynamical system, Combinatorial vector field, discrete Morse theory, isolated invariant set, Conley theory, isolating block..

Mathematics Subject Classification: Primary: 37B30; Secondary: 37B35, 37E15, 57M99, 57Q05, 57Q1.

Citation: Tomasz Kaczynski, Marian Mrozek, Thomas Wanner. Towards a formal tie between combinatorial and classical vector field dynamics. Journal of Computational Dynamics, 2016, 3 (1) : 17-50. doi: 10.3934/jcd.2016002

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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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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