# American Institute of Mathematical Sciences

October  2022, 42(10): 4787-4822. doi: 10.3934/dcds.2022072

## Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows

 1 Moscow State University, Faculty of Mechanics and Mathematics, Russia 2 Moscow Center for Fundamental and Applied Mathematics, Russia 3 Applied Mathematics and Physics Division, Gifu University, Yanagido 1-1, Gifu, 501-1193, Japan

*Corresponding author: Tomoo Yokoyama

Received  January 2022 Revised  April 2022 Published  October 2022 Early access  May 2022

Fund Project: VK is partially supported by Russian Science Foundation 21-11-00355 project. TY is partially supported by JSPS Kakenhi Grant Number 20K03583

It is known that $C^r$ Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any $r \in \mathbb{Z}_{\geq 1}$. In particular, $C^r$ Morse vector fields (i.e. Morse-Smale vector fields without limit cycles) form an open dense subset in the space of $C^r$ gradient vector fields on orientable closed surfaces and are structurally stable. Therefore generic time evaluations of gradient flows on orientable closed surfaces (e.g. solutions of differential equations) are described by alternating sequences of Morse flows and instantaneous non-Morse gradient flows. To illustrate the generic transitions (e.g. bifurcations of singular points, transitions via heteroclinic separatrices), we characterize and list all generic non-Morse gradient flows. To construct such characterizations, we characterize isolated singular points of gradient flows on surfaces. In fact, such a singular point is a non-trivial finitely sectored singular point without elliptic sectors. Moreover, considering Morse-Smale flows as "generic gradient flows with limit cycles", we characterize and list all generic non-Morse-Smale flows.

Citation: Vladislav Kibkalo, Tomoo Yokoyama. Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4787-4822. doi: 10.3934/dcds.2022072
##### References:
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##### References:
 [1] A. A. Andronov and L. S. Pontryagin, Rough systems, Dokl. Akad. Nauk SSSR, 14 (1937), 247-250. [2] S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, Translations of Mathematical Monographs, 153. American Mathematical Society, Providence, RI, 1996. doi: 10.1090/mmono/153. [3] C. Bonatti and R. Langevin, Difféomorphismes de Smale des Surfaces, Astérisque, 1998. [4] I. Bronstein and I. Nikolaev, Peixoto graphs of Morse-Smale foliations on surfaces, Topology Appl., 77 (1997), 19-36.  doi: 10.1016/S0166-8641(96)00101-0. [5] M. Cobo, C. Gutierrez and J. Llibre, Flows without wandering points on compact connected surfaces, Trans. Amer. Math. Soc., 362 (2010), 4569-4580.  doi: 10.1090/S0002-9947-10-05113-5. [6] V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma and O. V. Pochinka, Classification of Morse-Smale systems and topological structure of the underlying manifolds, Russian Mathematical Surveys, 74 (2019), 37-110.  doi: 10.4213/rm9855. [7] C. Gutiérrez, Structural stability for flows on the torus with a cross-cap, Trans. Amer. Math. Soc., 241 (1978), 311-320.  doi: 10.1090/S0002-9947-1978-0492303-2. [8] C. Gutierrez and B. Pires, On Peixoto's conjecture for flows on non-orientable 2-manifolds, Proc. Amer. Math. Soc., 133 (2005), 1063-1074.  doi: 10.1090/S0002-9939-04-07687-7. [9] C. Gutierrez and B. Pires, On $C^r$-closing for flows on orientable and non-orientable 2-manifolds, Bull. Braz. Math. Soc. (N.S.), 40 (2009), 553-576.  doi: 10.1007/s00574-009-0027-7. [10] S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjectures for flows, Ann. of Math., 145 (1997), 81-137.  doi: 10.2307/2951824. [11] G. Hector and U. Hirsch, Introduction to the Geometry of Foliations. Part A, Foliations on Compact Surfaces, Fundamentals for Arbitrary Codimension, and Holonomy, Second edition, Aspects of Mathematics, 1. Friedr. Vieweg & Sohn, Braunschweig, 1986. doi: 10.1007/978-3-322-90115-6. [12] V. Kruglov, D. Malyshev and O. Pochinka, Topological classification of $\Omega$-stable flows on surfaces by means of effectively distinguishable multigraphs, Discrete Contin. Dyn. Syst., 38 (2018), 4305-4327.  doi: 10.3934/dcds.2018188. [13] R. Labarca and M. J. Pacifico, Stability of Morse-Smale vector fields on manifolds with boundary, Topology, 29 (1990), 57-81.  doi: 10.1016/0040-9383(90)90025-F. [14] D. Malyshev, A. Morozov and O Pochinka, Combinatorial invariant for Morse-Smale diffeomorphisms on surfaces with orientable heteroclinic, Chaos, 31 (2021), 023119, 17 pp. doi: 10.1063/5.0029352. [15] N. G. Markley, On the number of recurrent orbit closures, Proc. Amer. Math. Soc., 25 (1970), 413-416.  doi: 10.1090/S0002-9939-1970-0256375-0. [16] A. Morozov and O. Pochinka, Morse-Smale surfaced diffeomorphisms with orientable heteroclinic, J. Dyn. Control Syst., 26 (2020), 629-639.  doi: 10.1007/s10883-019-09469-y. [17] I. Nikolaev, Foliations on Surfaces, A Series of Modern Surveys in Mathematics, 41. Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-04524-4. [18] I. Nikolaev and E. Zhuzhoma, Flows on 2-Dimensional Manifolds: An Overview, Lecture Notes in Mathematics, 1705. Springer-Verlag, Berlin, 1999. doi: 10.1007/BFb0093599. [19] A. A. Oshemkov and V. V. Sharko, Classification of Morse-Smale flows on two-dimensional manifolds, Sbornik: Mathematics, 189 (1998), 1205-1250.  doi: 10.1070/SM1998v189n08ABEH000341. [20] J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404.  doi: 10.1016/0040-9383(69)90024-X. [21] J. Palis and S. Smale, Structural stability theorems, 1970 Global Analysis, Amer. Math. Soc., Providence, R.I., 14 (1968), 223-231. [22] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120.  doi: 10.1016/0040-9383(65)90018-2. [23] C. Pugh and M. Shub, The $\Omega$-stability theorem for flows, Invent. Math., 11 (1970), 150-158.  doi: 10.1007/BF01404608. [24] C. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math., 89 (1967), 1010-1021.  doi: 10.2307/2373414. [25] C. C. Pugh and C. Robinson, The $C^1$ closing lemma, including hamiltonians, Ergodic Theory and Dynamical Systems, 3 (1983), 261-313.  doi: 10.1017/S0143385700001978. [26] S. Smale, On gradient dynamical systems, Annals of Mathematics, 74 (1961), 199-206.  doi: 10.2307/1970311. [27] X. Wang, The C*-algebras of Morse-Smale flows on two-manifolds, Ergodic Theory Dynam. Systems, 10 (1990), 565-597.  doi: 10.1017/S0143385700005757. [28] T. Yokoyama, Decompositions of surface flows, (2017), arXiv preprint, arXiv: 1703.05501. [29] T. Yokoyama, Refinements of topological invariants of flows, Discrete Contin. Dyn. Syst., 42 (2022), 2295-2331.  doi: 10.3934/dcds.2021191.
Two parabolic sectors $P^-$ and $P^+$, two hyperbolic sectors $H^-$ and $H^+$ with clockwise and anti-clockwise orbit directions, and two elliptic sectors $E^+$ and $E^-$ with clockwise and anti-clockwise orbit directions respectively
A saddle, two $\partial$-saddles, a sink, a $\partial$-sink, a source, a $\partial$-source, and a center
The resulting flow on the double of the Möbius band is a non-gradient Morse-like flow without limit circuits on a Klein bottle
The figure on the left is an orientable holonomy from $I$ to $J$ along a non-trivial circuit $\gamma$, and the figure on the right is a non-orientable holonomy from $I$ to $J$ along a non-trivial circuit $\gamma$
Canonical quotient mappings induced by the metric completion and the collapse for the case that $\Gamma$ consists of finitely many orbits
Fake parabolic separatrices
Left, all heteroclinic separatrices between a saddle/$\partial$-saddle $h_o$ and a saddle/$\partial$-saddle $h_i$ which are not contained in limit circuits of $h$-unstable flows; right, all pinching structures in $p$-unstable flows
Left, all homoclinic separatrices of saddles in $h$-unstable flows; right, all heteroclinic separatrices of $\partial$-saddles in limit circuits of $h$-unstable flows
A pair annihilation of an attracting limit cycle and a repelling limit cycle
An open flow box $U$ and a decomposition of its boundary
A perturbation of $\mu \subset \gamma$
Deformations of sectors to eliminate inner tangencies
Two successive tangencies on $\mu$
All flows in the figure are flows on a Möbius band with a $\partial$-sink, a $\partial$-source, two $\partial$-saddles, and non-recurrent orbits. The flow in the figure on the left is not Morse-Smale but regular Morse-like and quasi-Morse-Smale (see definition in §5) and has a non-trivial circuit with non-limit non-orientable holonomy. The flow in the second figure from the left is Morse. The flows in the second (resp. first) figures from the right are Morse-Smale and have a repelling (resp. attracting) limit cycle
Left, an orientable holonomy along a non-trivial circuit which is the closure of some homoclinic saddle separatrix $\gamma$; right, a non-orientable holonomy along a non-trivial circuit which is the closure of some homoclinic saddle separatrix $\gamma$
A loop which consists of finitely many non-degenerate orbit arcs and transverse arcs and its perturbed loop which consists of finitely many non-degenerate orbit arcs
Replacement of an orbit arc into a pair of two transverse arcs
Annihilation of a multi-saddle separatrix and creation of a topological limit cycle
Annihilation of a multi-saddle separatrix and creation of a topological limit cycle
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