Morse Predecomposition of an Invariant Set

• Michał Lipinski ^[3] ; Konstantin Mischaikow ^[1] ; Marian Mrozek ^[2]
  1. [1] Rutgers University

     Rutgers University

     City of New Brunswick, Estados Unidos

     
  2. [2] Jagiellonian University

     Jagiellonian University

     Kraków, Polonia

     
  3. [3] Polish Academy of Sciences (IMPAN) & Intitute of Science and Technology Austria (ISTA)

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 1, 2025
• Idioma: inglés
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• Resumen

  • Motivated by the study of recurrent orbits and dynamics within a Morse set of a Morse decomposition we introduce the concept of Morse predecomposition of an isolated invariant set within the setting of both combinatorial and classical dynamical systems. While Morse decomposition summarizes solely the gradient part of a dynamical system, the developed generalization extends to the recurrent component as well. In particular, a chain recurrent set, which is indecomposable in terms of Morse decomposition, can be represented more finely in the Morse predecomposition framework. This generalization is achieved by forgoing the poset structure inherent to Morse decomposition and relaxing the notion of connection between Morse sets (elements of Morse decomposition) in favor of what we term ’links’. We prove that a Morse decomposition is a special case of Morse predecomposition indexed by a poset. Additionally, we show how a Morse predecomposition may be condensed back to retrieve a Morse decomposition.

• Referencias bibliográficas
  • 1. Conley, C.: Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics, vol. 38. American Mathematical...
  • 2. Conley, C.: The gradient structure of a flow: I. Ergod. Th. & Dynam. Sys. 8∗, 11–26 (1988)
  • 3. Mrozek, M.: Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes. Found. Comput. Math. 17(6), 1585–1633...
  • 4. Lipi´nski, M., Kubica, J., Mrozek, M., Wanner, T.: Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite...
  • 5. Arai, Z., Kalies, W., Kokubu, H., Mischaikow, K., Oka, H., Pilarczyk, P.: A database schema for the analysis of global dynamics of multiparameter...
  • 6. Bush, J., Gameiro, M., Harker, S., Kokubu, H., Mischaikow, K., Obayashi, I., Pilarczyk, P.: Combinatorial-topological framework for the...
  • 7. Bush, J., Cowan, W., Harker, S., Mischaikow, K.: Conley-Morse databases for the angular dynamics of Newton’s method on the plane. SIAM...
  • 8. Liz, E., Pilarczyk, P.: Global dynamics in a stage-structured discrete-time population model with harvesting. J. Theoret. Biol. 297, 148–165...
  • 9. Mischaikow, K., Mrozek, M.: Isolating neighborhoods and chaos. Japan J. Indust. Appl. Math. 12(2), 205–236 (1995). https://doi.org/10.1007/BF03167289
  • 10. Szymczak, A.: The Conley index and symbolic dynamics. Topology 35(2), 287–299 (1996). https:// doi.org/10.1016/0040-9383(95)00029-1
  • 11. Day, S., Junge, O., Mischaikow, K.: A rigorous numerical method for the global analysis of infinitedimensional discrete dynamical systems....
  • 12. Day, S., Frongillo, R.: Sofic shifts via Conley index theory: computing lower bounds on recurrent dynamics for maps. SIAM J. Appl. Dyn....
  • 13. Pilarczyk, P., Signerska-Rynkowska, J., Graff, G.: Topological-numerical analysis of a twodimensional discrete neuron model. Chaos 33(4),...
  • 14. Woukeng, D., Sadowski, D., Leskiewicz, J., Lipi ´nski, M., Kapela, T.: Rigorous computation in dynamics based on topological methods for...
  • 15. Mrozek, M., Wanner, T.: Connection matrices in combinatorial topological dynamics. arXiv:2103.04269v2 (2023)
  • 16. Dey, T.K., Lipi ´nski, M., Mrozek, M., Slechta, R.: Computing connection matrices via persistence-like reductions. SIAM J. Appl. Dyn....
  • 17. Forman, R.: Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998)
  • 18. Forman, R.: Combinatorial vector fields and dynamical systems. Math. Z. 228(4), 629–681 (1998)
  • 19. Mrozek, M., Srzednicki, R., Thorpe, J., Wanner, T.: Combinatorial vs. classical dynamics: Recurrence. Commun. Nonlinear Sci. Numer. Simul....
  • 20. Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)
  • 21. Munkres, J.R.: Topology, p. 537. Prentice Hall Inc, Upper Saddle River, NJ (2000)
  • 22. Alexandrov, P.S.: Diskrete Räume. Mathematiceskii Sbornik (N.S.) 2, 501–518 (1937)
  • 23. Mischaikow, K., Mrozek, M.: Conley index theory. In: Handbook of Dynamical Systems. Handbook of Dynamical Systems, vol. 2, pp. 393–460....
  • 24. Akin, E.: The General Topology of Dynamical Systems, vol. 1. American Mathematical Society, Providence, RI (1993)
  • 25. Churchill, R.: Isolated invariant sets in compact metric spaces. J. Differential Equations 12(2), 330–352 (1972)
  • 26. Reineck, J.F.: The connection matrix in Morse–Smale flows. Trans. Amer. Math. Soc. 322(2), 523–545 (1990). https://doi.org/10.2307/2001713
  • 27. Franzosa, R.: Index filtrations and the homology index braid for partially ordered Morse decompositions. Trans. Amer. Math. Soc. 298(1),...
  • 28. Kalies, W.D., Mischaikow, K., Vandervorst, R.C.A.M.: Lattice structures for attractors III. J. Dynam. Differential Equations 34(3), 1729–1768...
  • 29. Franzosa, R.: Index Filtrations and Connection Matrices for Partially Ordered Morse Decompositions. PhD Thesis, University of Wisconsin-Madison...
  • 30. Williams, R.F.: The structure of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50, 73–99 (1979)
  • 31. Kaczynski, T., Mrozek, M., Wanner, T.: Towards a formal tie between combinatorial and classical vector field dynamics. J. Comput. Dyn....
  • 32. Silva, C.E.: Invitation to Ergodic Theory vol. 42. AMS Student Mathematical Library, (2008)
  • 33. Dey, T., Juda, M., Kapela, T., Kubica, J., Lipi ´nski, M., Mrozek, M.: Persistent homology of Morse decompositions in combinatorial dynamics....
  • 34. Mrozek, M., Wanner, T.: Creating semiflows on simplicial complexes from combinatorial vector fields. J. Diff. Equ. 304, 375–434 (2021)