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

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

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.
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]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[2]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[3]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[4]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[5]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[6]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[7]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[8]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[9]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[10]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020269

[11]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[12]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[13]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[14]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[15]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[16]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[17]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[18]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[19]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[20]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

 Impact Factor: 

Metrics

  • PDF downloads (110)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]