American Institute of Mathematical Sciences

Advanced
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

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.  doi: 10.1090/S0002-9947-99-02527-1.  Google Scholar

[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.  doi: 10.1137/15M1046691.  Google Scholar

[3]

C. Conley, Isolated Invariant Sets and the Morse Index,, American Mathematical Society, (1978).   Google Scholar

[4]

C. Conley and R. Easton, Isolated invariant sets and isolating blocks,, Transactions of the American Mathematical Society, 158 (1971), 35.  doi: 10.1090/S0002-9947-1971-0279830-1.  Google Scholar

[5]

R. Forman, Morse theory for cell complexes,, Advances in Mathematics, 134 (1998), 90.  doi: 10.1006/aima.1997.1650.  Google Scholar

[6]

R. Forman, Combinatorial vector fields and dynamical systems,, Mathematische Zeitschrift, 228 (1998), 629.  doi: 10.1007/PL00004638.  Google Scholar

[7]

L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, $2^{nd}$ ed,, Topological Fixed Point Theory and Its Applications, 4 (2006).   Google Scholar

[8]

T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology,, Applied Mathematical Sciences, 157 (2004).  doi: 10.1007/b97315.  Google Scholar

[9]

T. Kaczynski and M. Mrozek, Conley index for discrete multivalued dynamical systems,, Topology and Its Applications, 65 (1995), 83.  doi: 10.1016/0166-8641(94)00088-K.  Google Scholar

[10]

H. King, K. Knudson and N. Mramor, Generating discrete Morse functions from point data,, Experimental Mathematics, 14 (2005), 435.  doi: 10.1080/10586458.2005.10128941.  Google Scholar

[11]

M. Mrozek and B. Batko, Coreduction homology algorithm,, Discrete and Computational Geometry, 41 (2009), 96.  doi: 10.1007/s00454-008-9073-y.  Google Scholar

[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.  doi: 10.1109/TPAMI.2011.95.  Google Scholar

[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.  doi: 10.1137/140971075.  Google Scholar

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.  doi: 10.1090/S0002-9947-99-02527-1.  Google Scholar

[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.  doi: 10.1137/15M1046691.  Google Scholar

[3]

C. Conley, Isolated Invariant Sets and the Morse Index,, American Mathematical Society, (1978).   Google Scholar

[4]

C. Conley and R. Easton, Isolated invariant sets and isolating blocks,, Transactions of the American Mathematical Society, 158 (1971), 35.  doi: 10.1090/S0002-9947-1971-0279830-1.  Google Scholar

[5]

R. Forman, Morse theory for cell complexes,, Advances in Mathematics, 134 (1998), 90.  doi: 10.1006/aima.1997.1650.  Google Scholar

[6]

R. Forman, Combinatorial vector fields and dynamical systems,, Mathematische Zeitschrift, 228 (1998), 629.  doi: 10.1007/PL00004638.  Google Scholar

[7]

L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, $2^{nd}$ ed,, Topological Fixed Point Theory and Its Applications, 4 (2006).   Google Scholar

[8]

T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology,, Applied Mathematical Sciences, 157 (2004).  doi: 10.1007/b97315.  Google Scholar

[9]

T. Kaczynski and M. Mrozek, Conley index for discrete multivalued dynamical systems,, Topology and Its Applications, 65 (1995), 83.  doi: 10.1016/0166-8641(94)00088-K.  Google Scholar

[10]

H. King, K. Knudson and N. Mramor, Generating discrete Morse functions from point data,, Experimental Mathematics, 14 (2005), 435.  doi: 10.1080/10586458.2005.10128941.  Google Scholar

[11]

M. Mrozek and B. Batko, Coreduction homology algorithm,, Discrete and Computational Geometry, 41 (2009), 96.  doi: 10.1007/s00454-008-9073-y.  Google Scholar

[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.  doi: 10.1109/TPAMI.2011.95.  Google Scholar

[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.  doi: 10.1137/140971075.  Google Scholar

[1]

Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056
[2]

Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006
[3]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387
[4]

Bastian Laubner, Dierk Schleicher, Vlad Vicol. A combinatorial classification of postsingularly finite complex exponential maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 663-682. doi: 10.3934/dcds.2008.22.663
[5]

Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091
[6]

Joshua L. Mike, Vasileios Maroulas. Combinatorial Hodge theory for equitable kidney paired donation. Foundations of Data Science, 2019, 1 (1) : 87-101. doi: 10.3934/fods.2019004
[7]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[8]

Fabio Giannoni, Paolo Piccione, Daniel V. Tausk. Morse theory for the travel time brachistochrones in stationary spacetimes. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 697-724. doi: 10.3934/dcds.2002.8.697
[9]

Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629
[10]

J. B. van den Berg, J. D. Mireles James. Parameterization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4637-4664. doi: 10.3934/dcds.2016002
[11]

Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91
[12]

Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393
[13]

J. G. Ollason, N. Ren. A general dynamical theory of foraging in animals. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 713-720. doi: 10.3934/dcdsb.2004.4.713
[14]

Wei Liu, Shiji Song, Cheng Wu. Single-period inventory model with discrete stochastic demand based on prospect theory. Journal of Industrial & Management Optimization, 2012, 8 (3) : 577-590. doi: 10.3934/jimo.2012.8.577
[15]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693
[16]

Jijiang Sun, Shiwang Ma. Nontrivial solutions for Kirchhoff type equations via Morse theory. Communications on Pure & Applied Analysis, 2014, 13 (2) : 483-494. doi: 10.3934/cpaa.2014.13.483
[17]

Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 72-78. doi: 10.3934/proc.1998.1998.72
[18]

Chjan C. Lim. Extremal free energy in a simple mean field theory for a coupled Barotropic fluid - rotating sphere system. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 361-386. doi: 10.3934/dcds.2007.19.361
[19]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
[20]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure & Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249

 Impact Factor: 

Tools

Metrics

  • PDF downloads (102)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences