Classification of Morse–Smale systems and topological structure of the underlying manifolds

V. Z. Grines ; E. Ya. Gurevich ; E. V. Zhuzhoma ; O. V. Pochinka

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 1, pp. 37-110

Résumé

Morse–Smale systems arise naturally in applications for mathematical modelling of processes with regular dynamics (for example, in chains of coupled maps describing diffusion reactions, or in the study of the topology of magnetic fields in a conducting medium, in particular, in the study of the question of existence of separators in magnetic fields of highly conducting media). Since mathematical models in the form of Morse–Smale systems appear in the description of processes of various nature, the first step in the study of such models is to distinguish properties independent of the physical context but determining a partition of the phase space into trajectories. The relation preserving the partition into trajectories up to a homeomorphism is called topological equivalence, and the relation preserving also the time of motion along trajectories (continuous in the case of flows, and discrete in the case of cascades) is called topological conjugacy. The problem of topological classification of dynamical systems consists in finding invariants that uniquely determine the equivalence class or the conjugacy class for a given system. The present survey is devoted to a description of results on topological classification of Morse–Smale systems on closed manifolds, including results recently obtained by the authors. Also presented are recent results of the authors concerning the interconnections between the global dynamics of such systems and the topological structure of the underlying manifolds. Bibliography: 112 titles.

Détail

Keywords: Morse–Smale systems, topological classification, topology of the underlying manifold.

Bibliographie

[1] A. A. Andronov, L. S. Pontryagin, “Grubye sistemy”, Dokl. AN SSSR, 14:5 (1937), 247–250 | Zbl

[2] D. V. Anosov, “Roughness of geodesic flows on compact Riemannian manifolds of negative curvature”, Soviet Math. Dokl., 3 (1962), 1068–1070 | MR | Zbl

[3] D. V. Anosov, “Geodesic flows on closed Riemann manifolds with negative curvature”, Proc. Steklov Inst. Math., 90 (1967), 1–235 | MR | Zbl

[4] S. H. Aranson, “Trajectories on nonorientable two-dimensional manifolds”, Math. USSR-Sb., 9:3 (1969), 297–313 | DOI | MR | Zbl

[5] R. H. Fox, E. Artin, “Some wild cells and spheres in three-dimensional space”, Ann. of Math. (2), 49:4 (1948), 979–990 | DOI | MR | Zbl

[6] D. Asimov, “Round handles and non-singular Morse–Smale flows”, Ann. of Math. (2), 102:1 (1975), 41–54 | DOI | MR | Zbl

[7] D. Asimov, “Homotopy of non-singular vector fields to structurally stable ones”, Ann. of Math. (2), 102:1 (1975), 55–65 | DOI | MR | Zbl

[8] F. Béguin, “Classification des difféomorphismes de Smale des surfaces: types géométriques réalisables”, Ann. Inst. Fourier (Grenoble), 52:4 (2002), 1135–1185 | DOI | MR | Zbl

[9] F. Béguin, “Smale diffeomorphisms of surfaces: a classification algorithm”, Discrete Contin. Dyn. Syst., 11:2-3 (2004), 261–310 | DOI | MR | Zbl

[10] A. N. Bezdenezhnykh, V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. I, II”, Selecta Math. Soviet., 11:1 (1992), 1–11, 13–17 | MR | MR | MR | MR | Zbl

[11] A. N. Bezdenezhnykh, V. Z. Grines, “Realization of gradient-like diffeomorphisms of two-dimensional manifolds”, Selecta Math. Soviet., 11:1 (1992), 19–23 | MR | MR | Zbl

[12] C. Bonatti, V. Grines, “Knots as topological invariants for gradient-like diffeomorphisms of the sphere $S^3$”, J. Dynam. Control Systems, 6:4 (2000), 579–602 | DOI | MR | Zbl

[13] C. Bonatti, V. Grines, V. Medvedev, E. Pécou, “Three-manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves”, Topology Appl., 117:3 (2002), 335–344 | DOI | MR | Zbl

[14] C. Bonatti, V. Grines, V. Medvedev, E. Pécou, “Topological classification of gradient-like diffeomorphisms on 3-manifolds”, Topology, 43:2 (2004), 369–391 | DOI | MR | Zbl

[15] Ch. Bonatti, V. Z. Grines, O. V. Pochinka, “Classification of Morse–Smale diffeomorphisms with a finite set of heteroclinic orbits on 3-manifolds”, Proc. Steklov Inst. Math., 250 (2005), 1–46 | MR | Zbl

[16] Ch. Bonatti, V. Grines, O. Pochinka, “Classification of Morse–Smale diffeomorphisms with the chain of saddles on 3-manifolds”, Foliations 2005, World Sci. Publ., Hackensack, NJ, 2006, 121–147 | DOI | MR | Zbl

[17] Ch. Bonatti, V. Z. Grines, O. V. Pochinka, “Realization of Morse–Smale diffeomorphisms on $3$-manifolds”, Proc. Steklov Inst. Math., 297 (2017), 35–49 | DOI | DOI | MR | Zbl

[18] Ch. Bonatti, V. Grines, O. Pochinka, Topological classification of Morse–Smale diffeomorphisms on 3-manifolds, 2017, 48 pp., arXiv: 1710.08292

[19] Ch. Bonatti, V. Grines, O. Pochinka, S. Van Strien, A complete topological classification of Morse–Smale diffeomorphisms on surfaces: a kind of kneading theory in dimension two, preprint

[20] C. Bonatti, R. Langevin, Difféomorphismes de Smale des surfaces, Astérisque, 250, Soc. Math. France, Paris, 1998, viii+235 pp. | MR | Zbl

[21] Ch. Bonatti, L. Paoluzzi, “3-manifolds which are orbit spaces of diffeomorphisms”, Topology, 47:2 (2008), 71–100 | DOI | MR | Zbl

[22] M. Brown, “Locally flat imbeddings of topological manifolds”, Ann. of Math. (2), 75:2 (1962), 331–341 | DOI | MR | Zbl

[23] B. Campos, A. Cordero, J. Martínez-Alfaro, P. Vindel, “NMS flows on three-dimensional manifolds with one saddle periodic orbit”, Acta Math. Sin. (Engl. Ser.), 20:1 (2004), 47–56 | DOI | MR | Zbl

[24] B. Campos, J. Martínez-Alfaro, P. Vindel, “Bifurcations of links of periodic orbits in non-singular Morse–Smale systems on $S^3$”, Nonlinearity, 10:5 (1997), 1339–1355 | DOI | MR | Zbl

[25] B. Campos, P. Vindel, “Non equivalence of NMS flows on $S^3$”, Math. Bohemica, 137:2 (2012), 165–173 | MR | Zbl

[26] J. C. Cantrell, “Almost locally flat embeddings $S^{n-1}$ in $S^{n}$”, Bull. Amer. Math. Soc., 69:4 (1963), 716–718 | MR | Zbl

[27] J. C. Cantrell, C. H. Edwards, “Almost locally polyhedral curves in Euclidean $n$-space”, Trans. Amer. Math. Soc., 107:3 (1963), 451–457 | DOI | MR | Zbl

[28] A. Cobham, “The intrinsic computational difficulty of functions”, Logic, methodology, and philosophy of science, Proc. 1964 Internat. Congr., North-Holland, Amsterdam, 1965, 24–30 | MR | Zbl

[29] H. Debrunner, R. Fox, “A mildly wild imbedding of an $n$-frame”, Duke Math. J., 27:3 (1960), 425–429 | DOI | MR | Zbl

[30] J. Eells, Jr., N. H. Kuiper, “Manifolds which are like projective planes”, Inst. Hautes Études Sci. Publ. Math., 14 (1962), 5–46 | DOI | MR | Zbl

[31] G. Fleitas, “Classification of gradient-like flows on dimensions two and three”, Bol. Soc. Brasil. Mat., 6:2 (1975), 155–183 | DOI | MR | Zbl

[32] J. Franks, “The periodic structure of non-singular Morse–Smale flows”, Comment. Math. Helv., 53 (1978), 279–294 | DOI | MR | Zbl

[33] J. Franks, “Nonsingular Smale flows on $S^{3}$”, Topology, 24:3 (1985), 265–282 | DOI | MR | Zbl

[34] V. Z. Grines, “Topological classification of Morse–Smale diffeomorphisms with finite set of heteroclinic trajectories on surfaces”, Math. Notes, 54:3 (1993), 881–889 | DOI | MR | Zbl

[35] V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, “Peixoto graph of Morse–Smale diffeomorphisms on manifolds of dimension greater than three”, Proc. Steklov Inst. Math., 261 (2008), 59–83 | DOI | MR | Zbl

[36] V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, “Classification of Morse–Smale diffeomorphisms with one-dimensional set of unstable separatrices”, Proc. Steklov Inst. Math., 270 (2010), 57–79 | DOI | MR | Zbl

[37] V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, O. V. Pochinka, “On embedding a Morse–Smale diffeomorphism on a 3-manifold in a topological flow”, Sb. Math., 203:12 (2012), 1761–1784 | DOI | DOI | MR | Zbl

[38] V. Z. Grines, E. Ya. Gurevich, O. V. Pochinka, “The energy function of gradient-like flows and the topological classification problem”, Math. Notes, 96:6 (2014), 921–927 | DOI | DOI | MR | Zbl

[39] V. Z. Grines, E. Ya. Gurevich, O. V. Pochinka, “Topological classification of Morse–Smale diffeomorphisms without heteroclinic intersections”, J. Math. Sci. (N. Y.), 208:1 (2015), 81–90 | DOI | MR | Zbl

[40] V. Z. Grines, E. Ya. Gurevich, O. V. Pochinka, “Kombinatornyi invariant dlya kaskadov Morsa–Smeila bez geteroklinicheskikh peresechenii na sfere $S^n$, $n\geqslant 4$”, Matem. zametki, 105:1 (2019), 136–141 | DOI

[41] V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, V. S. Medvedev, “O topologii mnogoobrazii, dopuskayuschikh gradientno-podobnye potoki s zadannym nebluzhdayuschim mnozhestvom”, Matem. trudy, 21:2 (2018), 163–180 | DOI

[42] V. Z. Grines, S. Kh. Kapkaeva, O. V. Pochinka, “A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces”, Sb. Math., 205:10 (2014), 1387–1412 | DOI | DOI | MR | Zbl

[43] V. Z. Grines, D. S. Malyshev, O. V. Pochinka, S. Kh. Zinina, “Efficient algorithms for the recognition of topologically conjugate gradient-like diffeomorphisms”, Regul. Chaotic Dyn., 21:2 (2016), 189–203 | DOI | MR | Zbl

[44] V. Z. Grines, T. V. Medvedev, O. V. Pochinka, Dynamical systems on 2- and 3-manifolds, Dev. Math., 46, Springer, Cham, 2016, xxvi+295 pp. | DOI | MR | Zbl

[45] V. Grines, T. Medvedev, O. Pochinka, E. Zhuzhoma, “On heteroclinic separators of magnetic fields in electrically conducting fluids”, Phys. D, 294 (2015), 1–5 | DOI | MR | Zbl

[46] V. Z. Grines, T. M. Mitryakova, O. V. Pochinka, “Novye topologicheskie invarianty negradientno-podobnykh diffeomorfizmov na orientiruemykh poverkhnostyakh”, Tr. Srednevolzhskogo matem. o-va, 7:1 (2005), 123–129

[47] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, “New relations for Morse–Smale systems with trivially embedded one-dimensional separatrices”, Sb. Math., 194:7 (2003), 979–1007 | DOI | DOI | MR | Zbl

[48] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, “On Morse–Smale diffeomorphisms with four periodic points on closed orientable manifolds”, Math. Notes, 74:3 (2003), 352–366 | DOI | DOI | MR | Zbl

[49] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, “On the structure of the ambient manifold for Morse–Smale systems without heteroclinic intersections”, Proc. Steklov Inst. Math., 297 (2017), 179–187 | DOI | DOI | MR | Zbl

[50] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms”, Proc. Steklov Inst. Math., 271 (2010), 103–124 | DOI | MR | Zbl

[51] V. Z. Grines, E. V. Zhuzhoma, O. V. Pochinka, “Sistemy Morsa–Smeila i topologicheskaya struktura nesuschikh mnogoobrazii”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 61, RUDN, M., 2016, 5–40 | MR

[52] V. Z. Grines, E. V. Zhuzhoma, O. V. Pochinka, “Dinamicheskie sistemy i topologiya magnitnykh polei v provodyaschei srede”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 3, RUDN, M., 2017, 455–474 | MR

[53] D. M. Grobman, “O gomeomorfizme sistem differentsialnykh uravnenii”, Dokl. AN SSSR, 128:5 (1959), 880–881 | MR | Zbl

[54] D. M. Grobman, “Topologicheskaya klassifikatsiya okrestnostei osoboi tochki v $n$-mernom prostranstve”, Matem. sb., 56(98):1 (1962), 77–94 | MR | Zbl

[55] E. Ya. Gurevich, V. S. Medvedev, “O mnogoobraziyakh razmernosti $n$, dopuskayuschikh diffeomorfizmy s sedlovymi tochkami indeksov $1$ i $n-1$”, Tr. Crednevolzhskogo matem. o-va, 8:1 (2006), 204–208 | Zbl

[56] C. Gutiérrez, “Structural stability for flows on the torus with a cross-cap”, Trans. Amer. Math. Soc., 241 (1978), 311–320 | DOI | MR | Zbl

[57] P. Hartman, “On the local linearization of differential equations”, Proc. Amer. Math. Soc., 14:4 (1963), 568–573 | DOI | MR | Zbl

[58] M. W. Hirsch, C. C. Pugh, M. Shub, Invariant manifolds, Lecture Notes in Math., 583, Springer-Verlag, Berlin–New York, 1977, ii+149 pp. | DOI | MR | Zbl

[59] M. A. Kervaire, J. W. Milnor, “Groups of homotopy spheres. I”, Ann. of Math. (2), 77:3 (1963), 504–537 | DOI | MR | Zbl

[60] R. C. Kirby, “On the set of non-locally flat points of a submanifold of codimension one”, Ann. of Math. (2), 88:2 (1968), 281–290 | DOI | MR | Zbl

[61] V. Kruglov, D. Malyshev, O. Pochinka, “Topological classification of $\Omega$-stable flows on surfaces by means of effectively distinguishable multigraphs”, Discrete Contin. Dyn. Syst., 38:9 (2018), 4305–4327 | DOI | MR | Zbl

[62] V. E. Kruglov, D. S. Malyshev, O. V. Pochinka, “On algorithms that effectively distinguish gradient-like dynamics”, Regul. Chaotic Dyn. (to appear)

[63] R. Langevin, “Quelques nouveaux invariants des difféomorphismes Morse–Smale d'une surface”, Ann. Inst. Fourier (Grenoble), 43:1 (1993), 265–278 | DOI | MR | Zbl

[64] J. M. Lee, Introduction to smooth manifolds, Grad. Texts in Math., 218, 2nd rev. ed., Spinger, New York, 2013, xvi+708 pp. | DOI | MR | Zbl

[65] E. A. Leontovich, A. G. Maier, “O traektoriyakh, opredelyayuschikh kachestvennuyu strukturu razbieniya sfery na traektorii”, Dokl. AN SSSR, 14:5 (1937), 251–257 | Zbl

[66] E. A. Leontovich, A. G. Maier, “O skheme, opredelyayuschei topologicheskuyu strukturu razbieniya na traektorii”, Dokl. AN SSSR, 103:4 (1955), 557–560 | MR | Zbl

[67] A. G. Maier, “Gruboe preobrazovanie okruzhnosti v okruzhnost”, Uch. zap. Gork. gos. un-ta, 1939, no. 12, 215–229

[68] N. G. Markley, “The Poincaré–Bendixson theorem for the Klein bottle”, Trans. Amer. Math. Soc., 135 (1969), 159–165 | DOI | MR | Zbl

[69] Y. Matsumoto, An introduction to Morse theory, Transl. from the Japan., Transl. Math. Monogr., 208, Iwanami Series in Modern Mathematics, Amer. Math. Soc., Providence, RI, 2002, xiv+219 pp. | MR | Zbl

[70] S. V. Matveev, “Classification of sufficiently large three-dimensional manifolds”, Russian Math. Surveys, 52:5 (1997), 1029–1055 | DOI | DOI | MR | Zbl

[71] V. S. Medvedev, E. V. Zhuzhoma, “Morse–Smale systems with few non-wandering points”, Topology Appl., 160:3 (2013), 498–507 | DOI | MR | Zbl

[72] K. R. Meyer, “Energy functions for Morse Smale systems”, Amer. J. Math., 90:4 (1968), 1031–1040 | DOI | MR | Zbl

[73] J. Milnor, “On manifolds homeomorphic to the 7-sphere”, Ann. of Math. (2), 64:2 (1956), 399–405 | DOI | MR | Zbl

[74] J. Milnor, Morse theory, Ann. of Math. Stud., 51, Princeton Univ. Press, Princeton, NJ, 1963, vi+153 pp. | MR | Zbl

[75] T. M. Mitryakova, O. V. Pochinka, “Necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency”, Proc. Steklov Inst. Math., 270 (2010), 194–215 | DOI | MR | Zbl

[76] J. W. Morgan, “Non-singular Morse–Smale flows on 3-dimensional manifolds”, Topology, 18:1 (1979), 41–53 | DOI | MR | Zbl

[77] M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ., 18, Amer. Math. Soc., New York, 1934, xii+368 pp. | MR | Zbl

[78] J. Nielsen, “Die Structur periodischer Transformationen von Flächen”, Danske Vidensk. Selsk. Math.-Fys. Medd., 15:1 (1937), 1–77 | Zbl

[79] I. Nikolaev, “Graphs and flows on surfaces”, Ergodic Theory Dynam. Systems, 18:1 (1998), 207–220 | DOI | MR | Zbl

[80] I. Nikolaev, E. Zhuzhoma, Flows on 2-dimensional manifolds. An overview, Lecture Notes in Math., 1705, Springer-Verlag, Berlin, 1999, xx+294 pp. | DOI | MR | Zbl

[81] A. A. Oshemkov, V. V. Sharko, “Classification of Morse–Smale flows on two-dimensional manifolds”, Sb. Math., 189:8 (1998), 1205–1250 | DOI | DOI | MR | Zbl

[82] J. Palis, “On Morse–Smale dynamical systems”, Topology, 8:4 (1969), 385–404 | DOI | MR | Zbl

[83] J. Palis, Jr., W. de Melo, Geometric theory of dynamical systems. An introduction, Transl. from the Portuguese, Springer-Verlag, New York–Berlin, 1982, xii+198 pp. | MR | MR | Zbl

[84] J. Palis, S. Smale, “Structural stability theorems”, Global analysis (Berkeley, CA, 1968), Proc. Sympos. Pure Math., 14, Amer. Math. Soc., Providence, RI, 1970, 223–231 | DOI | MR | Zbl

[85] M. M. Peixoto, “On structural stability”, Ann. of Math. (2), 69 (1959), 199–222 | DOI | MR | Zbl

[86] M. M. Peixoto, “Structural stability on two-dimensional manifolds”, Topology, 1:2 (1962), 101–120 ; A further remark, Topology, 2:1-2 (1963), 179–180 | DOI | MR | Zbl | DOI | MR | Zbl

[87] M. M. Peixoto, “On the classification of flows on 2-manifolds”, Dynamical systems, Proc. Symp. Dyn. Syst. (Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 389–419 | DOI | MR | Zbl

[88] Ya. B. Pesin, A. A. Yurchenko, “Some physical models of the reaction-diffusion equation, and coupled map lattices”, Russian Math. Surveys, 59:3 (2004), 481–513 | DOI | DOI | MR | Zbl

[89] S. Ju. Piljugin, “Phase diagrams determining Morse–Smale systems without periodic trajectories on spheres”, Differential Equations, 14 (1978), 170–177 | MR | Zbl

[90] D. Pixton, “Wild unstable manifolds”, Topology, 16:2 (1977), 167–172 | DOI | MR | Zbl

[91] V. A. Pliss, “O grubosti differentsialnykh uravnenii, zadannykh na tore”, Vestn. Leningrad. un-ta. Ser. 1. Matem. Mekh. Astron., 13 (1960), 15–23 | MR | Zbl

[92] A. O. Prishlyak, “Morse–Smale vector fields without closed trajectories on 3-manifolds”, Math. Notes, 71:2 (2002), 230–235 | DOI | DOI | MR | Zbl

[93] A. Prishlyak, “Complete topological invariants of Morse–Smale flows and handle decompositions of 3-manifolds”, J. Math. Sci. (N. Y.), 144:5 (2007), 4492–4499 | DOI | MR | Zbl

[94] C. Pugh, M. Shub, “The $\Omega$-stability theorem for flows”, Invent. Math., 11:2 (1970), 150–158 | DOI | MR | Zbl

[95] C. C. Pugh, R. B. Walker, F. W. Wilson, “On Morse–Smale approximations – a counterexample”, J. Differential Equations, 23:1 (1977), 173–182 | DOI | MR | Zbl

[96] T. Radó, “Über den Begriff der Riemannschen Fläche”, Acta Univ. Szeged, 2 (1925), 101–121 | Zbl

[97] G. Zeifert, V. Trelfall, Topologiya, GONTI, M.–L., 1938, 400 pp.; H. Seifert, W. Threlfall, Lehrbuch der Topologie, B. G. Teubner, Leipzig, 1934, iv+353 pp. ; H. Seifert, W. Threlfall, A textbook of topology, Pure Appl. Math., 89, Academic Press, Inc., New York–London, 1980, xvi+437 СЃ. | Zbl | MR | Zbl

[98] M. Shub, D. Sullivan, “Homology theory and dynamical systems”, Topology, 14:2 (1975), 109–132 | DOI | MR | Zbl

[99] S. Smale, “Morse inequalities for a dynamical system”, Bull. Amer. Math. Soc., 66 (1960), 43–49 | DOI | MR | Zbl

[100] S. Smale, “On gradient dynamical systems”, Ann. of Math. (2), 74:1 (1961), 199–206 | DOI | MR | Zbl

[101] S. Smale, “Diffeomorphisms with many periodic points”, Differential and combinatorial topology, Princeton Univ. Press, Princeton, NJ, 1965, 63–80 | MR | Zbl

[102] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817 | DOI | MR | MR | Zbl

[103] Ya. L. Umanskiĭ, “Necessary and sufficient conditions for topological equivalence of three-dimensional Morse–Smale dynamical systems with a finite number of singular trajectories”, Math. USSR-Sb., 69:1 (1991), 227–253 | DOI | MR | Zbl

[104] I. Yu. Vlasenko, “A full invariant for Morse–Smale diffeomorphisms on non-orientable surfaces”, Russian Math. Surveys, 54:5 (1999), 1058–1059 | DOI | DOI | MR | Zbl

[105] M. Wada, “Closed orbits of non-singular Morse–Smale flows on $S^3$”, J. Math. Soc. Japan, 41:3 (1989), 405–413 | DOI | MR | Zbl

[106] F. Waldhausen, “On irreducible 3-manifolds which are sufficiently large”, Ann. of Math. (2), 87 (1968), 56–88 | DOI | MR | Zbl

[107] X. Wang, “The $C^*$-algebras of Morse–Smale flows on two-manifolds”, Ergodic Theory Dynam. Systems, 10:3 (1990), 565–597 | DOI | MR | Zbl

[108] K. Yokoyama, “Classification of periodic maps on compact surfaces. I”, Tokyo J. Math., 6:1 (1983), 75–94 | DOI | MR | Zbl

[109] B. Yu, “Behavior 0 nonsingular Morse Smale flows on $S^3$”, Discrete Contin. Dyn. Syst., 36:1 (2016), 509–540 | DOI | MR | Zbl

[110] E. V. Zhuzhoma, V. S. Medvedev, “Gradient flows with wildly embedded closures of separatrices”, Proc. Steklov Inst. Math., 270 (2010), 132–140 | DOI | MR | Zbl

[111] E. V. Zhuzhoma, V. S. Medvedev, “Morse–Smale systems with three nonwandering points”, Dokl. Math., 84:2 (2011), 604–606 | DOI | MR | Zbl

[112] E. V. Zhuzhoma, V. S. Medvedev, “Continuous Morse–Smale flows with three equilibrium positions”, Sb. Math., 207:5 (2016), 702–723 | DOI | DOI | MR | Zbl