-
Previous Article
Discretization strategies for computing Conley indices and Morse decompositions of flows
- JCD Home
- This Issue
-
Next Article
On the numerical approximation of the Perron-Frobenius and Koopman operator
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.
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.
doi: 10.1137/15M1046691. |
[3] |
C. Conley, Isolated Invariant Sets and the Morse Index,, American Mathematical Society, (1978).
|
[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. |
[5] |
R. Forman, Morse theory for cell complexes,, Advances in Mathematics, 134 (1998), 90.
doi: 10.1006/aima.1997.1650. |
[6] |
R. Forman, Combinatorial vector fields and dynamical systems,, Mathematische Zeitschrift, 228 (1998), 629.
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 (2006).
|
[8] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology,, Applied Mathematical Sciences, 157 (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.
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.
doi: 10.1080/10586458.2005.10128941. |
[11] |
M. Mrozek and B. Batko, Coreduction homology algorithm,, Discrete and Computational Geometry, 41 (2009), 96.
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.
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.
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.
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.
doi: 10.1137/15M1046691. |
[3] |
C. Conley, Isolated Invariant Sets and the Morse Index,, American Mathematical Society, (1978).
|
[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. |
[5] |
R. Forman, Morse theory for cell complexes,, Advances in Mathematics, 134 (1998), 90.
doi: 10.1006/aima.1997.1650. |
[6] |
R. Forman, Combinatorial vector fields and dynamical systems,, Mathematische Zeitschrift, 228 (1998), 629.
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 (2006).
|
[8] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology,, Applied Mathematical Sciences, 157 (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.
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.
doi: 10.1080/10586458.2005.10128941. |
[11] |
M. Mrozek and B. Batko, Coreduction homology algorithm,, Discrete and Computational Geometry, 41 (2009), 96.
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.
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.
doi: 10.1137/140971075. |
[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] |
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 |
[7] |
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 |
[8] |
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 |
[9] |
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 |
[10] |
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 |
[11] |
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 |
[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] |
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 |
[17] |
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 |
[18] |
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 |
[19] |
Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483 |
[20] |
Valery Imaikin, Alexander Komech, Herbert Spohn. Scattering theory for a particle coupled to a scalar field. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 387-396. doi: 10.3934/dcds.2004.10.387 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]