Minimal flow morsification subject to level set control: a combinatorial approach
DOI:
https://doi.org/10.12775/TMNA.2024.049Słowa kluczowe
Lyapunov graph, Morse-Conley theory, Poincaré-Hopf inequalities, network-flow theoryAbstrakt
In this paper, we address both a combinatorial continuation question via Lyapunov graphs and the attainability of a preassigned level set, dubbed ground level set and given by its Betti numbers, within a morsification process of a dynamical configuration. The novelty introduced here is a componentwise Lyapunov graph morsification that keeps track of level sets during the morsification process subject to the minimality of the total number of singularities of a morsified flow. The algorithm behind the morsification translates into a system of linear equations whose feasibility is linked to that of a set of inequalities, called componentwise Poincaré-Hopf inequalities, involving the input data. This investigation combines techniques from both homological Conley index and network flow theories.Bibliografia
R.K. Ahuja, T.L. Magnanti and J.B. Orlin, Network Flows: Theory, Applications and Algorithms, Prentice-Hall, Englewood Cliffs, New Jersey, USA, 1993.
M. Bertolim, K. de Rezende and O. Manzoli Neto, Isolating blocks for periodic orbits, J. Dynam. Control Systems 13 (2007), no. 1, 121–134.
M. Bertolim, K. de Rezende, O. Manzoli Neto and G. Vago, Isolating blocks for Morse flows, Geom. Dedicata 121 (2006), 19–41.
M. Bertolim, K. De Rezende and M. Mello, Bounds on the Ogasa number for ordered continuations of Lyapunov graphs, JP Journal of Geometry and Topology 13 (2023), no. 1, 41–91.
M. Bertolim, K. de Rezende, and G. Vago, Minimal Morse flows on compact manifolds, Topology Appl. 153 (2006), no. 18, 3450–3466.
M. Bertolim, M. Mello and K. de Rezende, Dynamical and topological aspects of Lyapunov graphs, Qual. Theory Dyn. Syst. 4 (2003), no. 2, 181–203.
M. Bertolim, M. Mello and K. de Rezende, Lyapunov graph continuation, Ergodic Theory Dynam. Systems 23 (2003), no. 1, 1–58.
M. Bertolim, M. Mello and K. de Rezende, Poincaré-Hopf and Morse inequalities for Lyapunov graphs, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 1–39.
M. Bertolim, M. Mello and K. de Rezende, Poincaré-Hopf inequalities, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4091–4129 (electronic).
M.A. Bertolim, C. Bonatti, M.P. Mello and G.M. Vago, Minimal number of periodic orbits for non-singular Morse–Smale flows in odd dimension, Fund. Math. 267 (2024), no. 1, 25–85.
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978.
R. Cruz and K. de Rezende, Gradient-like flows on high-dimensional manifolds, Ergodic Theory Dynam. Systems 19 (1999), no. 2, 339–362.
R.N. Cruz and K.A. de Rezende, Cycle rank of Lyapunov graphs and the genera of manifolds, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3715–3720.
G.B. Dantzig and M.N. Thapa, Linear Programming 1, Springer Series in Operations Research and Financial Engineering, Springer, New York, NY, 1997.
K.A. De Rezende, Smale Flows on the three-sphere (Dynamical Systems, Lyapunov Graph), ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.), Northwestern University, 1985.
K.A. de Rezende, G.G.E. Ledesma, O. Manzoli-Neto and G.M. Vago, Lyapunov graphs for circle valued functions, Topology Appl. 245 (2018), 62–91.
J. Franks, Nonsingular Smale flows on S 3 , Topology 24 (1985), no. 3, 265–282.
I. Gelbukh, Loops in Reeb graphs of n-manifolds, Discrete Comput. Geom. 59 (2018), no. 4, 843–863.
I. Gelbukh, Approximation of metric spaces by Reeb graphs: cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifolds, Filomat33 (2019), no. 7, 2031–2049.
I. Gelbukh, Realization of a digraph as the Reeb graph of a Morse–Bott function on a given surface, Topology Appl. 341 (2024), paper no. 108745, 15.
F. Gibou, R. Fedkiw and S. Osher, A review of level-set methods and some recent applications, J. Comput. Phys. 353 (2018), 82–109.
G.G.E. Ledesma, D.V.S. Lima, M.P. Mello, K.A. de Rezende and M.R. da Silveira, Homotopical dynamics for gradient-like flows, 34◦ Colóquio Brasileiro de Matemática, vol. 10, Editora do IMPA, Rio de Janeiro, 1 edition, 2023.
D.V. de S. Lima, O.M. Neto, K.A. de Rezende and M.R. da Silveira, Cancellations for circle-valued Morse functions via spectral sequences, Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 259–311.
L.P. Michalak, Realization of a graph as the Reeb graph of a Morse function on a manifold, Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 749–762.
E. Ogasa, A new invariant associated with decompositions of manifolds (2013), https://arxiv.org/pdf/math/0512320v5.
G. Reeb, Sur les points singuliers dune forme de Pfaff complètement intégrable ou dune fonction numérique, C.R. Acad. Sci. Paris 222 (1946), 847–849.
J.A. Sethian and J. Schwartz, Theory, algorithms, and applications of level set methods for propagating interfaces, Acta Numerica 5 (1996), 309–395.
R. Tsai and S. Osher, Level set methods and their applications in image science, Commun. Math. Sci. 1 (2003), no. 4, 623–656.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
