Geodesic
On some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive dynamical systems)
Sbornik. Mathematics, Tome 19 (1973) no. 3, pp. 365-393 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, topological invariants of dynamical systems given on a two-dimensional manifold $M^2$ of genus $p>1$ are selected which allow one to distinguish topologically inequivalent systems which have nonclosed, Poisson stable trajectories and non-null-homotopic closed trajectories. A necessary and sufficient condition for the topological equivalence of transitive dynamical systems on $M^2$ is established. Figures: 6. Bibliography: 20 titles. 
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S. Kh. Aranson; V. Z. Grines. On some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive dynamical systems). Sbornik. Mathematics, Tome 19 (1973) no. 3, pp. 365-393. http://geodesic.mathdoc.fr/item/SM_1973_19_3_a1/

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

[2] A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Kachestvennaya teoriya dinamicheskikh sistem vtorogo poryadka, izd-vo «Nauka», Moskva, 1966 | MR

[3] A. Puankare, O krivykh, opredelyaemykh differentsialnymi uravneniyami, Gostekhizdat, Moskva–Leningrad, 1947

[4] H. Kneser, “Reguläre Kurvenscharen auf Ringflächen”, Math. Ann., 91 (1923), 135–154 | DOI | MR

[5] V. V. Nemytskii, V. V. Stepanov, Kachestvennaya teoriya differentsialnykh uravnenii, Gostekhizdat, Moskva–Leningrad

[6] A. Aeppli, L. Markus, “Integral equivalence of vector fields on manifolds and bifurcation of differential systems”, Amer. J. Math., 85:4 (1963), 633–654 | DOI | MR | Zbl

[7] N. G. Markley, “Homeomorphisms of the circle without periodic points”, J. London Math. Soc., 20 (1970), 688–698 | DOI | MR | Zbl

[8] L. E. Reizin, “Topologicheskaya klassifikatsiya dinamicheskikh sistem bez tochek pokoya na tore”, Latv. matem. ezhegodnik, 5 (1969), 113–121 | MR | Zbl

[9] A. G. Maier, “O traektoriyakh na orientiruemykh poverkhnostyakh”, Matem. sb., 12 (54) (1943), 71–84

[10] S. Kh. Aranson, “Ob otsutstvii nezamknutykh ustoichivykh po Puassonu polutraektorii i traektorii, dvoyakoasimptoticheskikh k dvoinomu predelnomu tsiklu, u dinamicheskikh sistem pervoi stepeni negrubosti na orientiruemykh dvumernykh mnogoobraziyakh”, Matem. sb., 76 (118) (1968), 214–230 | MR | Zbl

[11] S. Kh. Aranson, “Traektorii na neorientiruemykh dvumernykh mnogoobraziyakh”, Matem. sb., 80 (122) (1969), 314–333 | MR | Zbl

[12] S. Kh. Aranson, “Dinamicheskie sistemy na dvumernykh mnogoobraziyakh”, Trudy V mezhdunar. konferents. po nelineinym kolebaniyam, t. 2, Kiev, 1970, 46–52 | Zbl

[13] L. R. Ford, Avtomorfnye funktsii, ONTI, Moskva–Leningrad, 1936

[14] D. Springer, Vvedenie v teoriyu rimanovykh poverkhnostei, IL, Moskva, 1960

[15] J. Nielsen, “Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen”, Acta math., 50 (1927), 189–358 | DOI | MR | Zbl

[16] L. S. Pontryagin, “Nekotorye topologicheskie invarianty zamknutykh rimanovykh mnogoobrazii”, Izv. AN SSSR, seriya matem., 13 (1949), 125–162 | MR | Zbl

[17] V. I. Pupko, “O nesamoperesekayuschikhsya krivykh na zamknutykh poverkhnostyakh”, DAN SSSR, 177:2 (1967), 272–274 | MR | Zbl

[18] T. M. Cherry, “Topological properties of the solutions of ordinary differential equations”, Amer. J. Math., 59 (1937), 957–982 | DOI | MR | Zbl

[19] J. Nielsen, “Über topologische Abbildungen geschlossener Flächen”, Abhandl. math. Seminar Hamburg. Univ., III:1 (1924), 246–260 | DOI

[20] R. J. Sacker, G. R. Sell, “On the existence of periodic solutions on 2-manifolds”, J. diff. equations, 11:3 (1972), 449–463 | DOI | MR | Zbl