The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

Krzysztof Ciesielski

Open Mathematics, Tome 10 (2012), p. 2110-2128 / Harvested from The Polish Digital Mathematics Library

Résumé

The Poincaré-Bendixson Theorem and the development of the theory are presented - from the papers of Poincaré and Bendixson to modern results.

Détail

Publié le : 2012-01-01

Zbl 06137115

EUDML-ID : urn:eudml:doc:269766

Bibliographie

[1] Aarts J.M., de Vries J., Colloquium Topologische Dynamische Systemen, MC Syllabus, 36, Mathematisch Centrum, Amsterdam, 1977

[2] Abate M., Tovena F., Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields, J. Differential Equations, 2011, 251(9), 2612–2684 http://dx.doi.org/10.1016/j.jde.2011.05.031[Crossref] | Zbl 1241.32012

[3] Arrowsmith D.K., Place C.M., Ordinary Differential Equations, Chapman and Hall Math. Ser., Chapman & Hall, London, 1982 | Zbl 0481.34005

[4] Athanassopoulos K., Flows with cyclic winding number groups, J. Reine Angew. Math., 1996, 481, 207–215 | Zbl 0858.58041

[5] Athanassopoulos K., One-dimensional chain recurrent sets of flows in the 2-sphere, Math. Z., 1996, 223(4), 643–649 http://dx.doi.org/10.1007/PL00004279[Crossref] | Zbl 0872.58051

[6] Athanassopoulos K., Petrescou T., Strantzalos P., A class of flows on 2-manifolds with simple recurrence, Comment. Math. Helv., 1997, 72(4), 618–635 http://dx.doi.org/10.1007/s000140050038[Crossref] | Zbl 0927.37023

[7] Athanassopoulos K., Strantzalos P., On minimal sets in 2-manifolds, J. Reine Angew. Math., 1988, 388, 206–211 | Zbl 0647.58027

[8] Balibrea F., Jiménez López V., A characterization of the ω-limit sets of planar continuous dynamical systems, J. Differential Equations, 1998, 145(2), 469–488 http://dx.doi.org/10.1006/jdeq.1997.3401[Crossref]

[9] Bebutoff, M. Sur la représentation des trajectoires d’un système dynamique sur un système de droites parallèles, Bull. Math. Univ. Moscou, 1939, 2(3), 1–22

[10] Beck A., Continuous Flows in the Plane, Grundlehren Math. Wiss., 201, Springer, New York-Heidelberg, 1974 http://dx.doi.org/10.1007/978-3-642-65548-7[Crossref] | Zbl 0295.54001

[11] Bendixson I., Sur les courbes définies par des équations différentielles, Acta Math., 1901, 24(1), 1–88 http://dx.doi.org/10.1007/BF02403068[Crossref] | Zbl 31.0328.03

[12] Bhatia N.P., Attraction and nonsaddle sets in dynamical systems, J. Differential Equations, 1970, 8, 229–249 http://dx.doi.org/10.1016/0022-0396(70)90003-3[Crossref] | Zbl 0207.08803

[13] Bhatia N.P., Hájek O., Local Semi-Dynamical Systems, Lecture Notes in Math., 90, Springer, Berlin-New York, 1969 | Zbl 0176.39102

[14] Bhatia N.P., Lazer A.C., Leighton W., Application of the Poincaré-Bendixson Theorem, Ann. Mat. Pura Appl., 1966, 73, 27–32 http://dx.doi.org/10.1007/BF02415080[Crossref] | Zbl 0144.22403

[15] Bhatia N.P., Szegő G.P., Stability Theory of Dynamical Systems, Grundlehren Math. Wiss., 161, Springer, New York-Berlin, 1970 http://dx.doi.org/10.1007/978-3-642-62006-5[Crossref] | Zbl 0213.10904

[16] Birkhoff G.D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., 9, American Mathematical Society, Providence, 1927 | Zbl 53.0732.01

[17] Bohr H., Fenchel W., Ein Satz über stabile Bewegungen in der Ebene, Danske Vid. Selsk. Mat.-Fys. Medd., 1936, 14(1), 1–15 | Zbl 62.0919.02

[18] Bonotto E.M., Federson M., Limit sets and the Poincaré-Bendixson theorem in impulsive semidynamical systems, J. Differential Equations, 2008, 244(9), 2334–2349 http://dx.doi.org/10.1016/j.jde.2008.02.007[Crossref] | Zbl 1143.37014

[19] Chewning W.C., A dynamical system on E 4 neither isomorphic nor equivalent to a differentiable system, Bull. Amer. Math. Soc., 1974, 80, 150–153 http://dx.doi.org/10.1090/S0002-9904-1974-13396-3 | Zbl 0273.54029

[20] Ciesielski K., Sections in semidynamical systems, Bull. Polish Acad. Sci. Math., 1992, 40, 61–70 | Zbl 0602.54040

[21] Ciesielski K., The Poincaré-Bendixson theorems for two-dimensional semiflows, Topol. Methods Nonlinear Anal., 1994, 3(1), 163–178 | Zbl 0839.54028

[22] Coddington E.A., Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill, New York-Toronto-London, 1955 | Zbl 0064.33002

[23] Cronin J., Differential Equations, Monographs and Textbooks in Pure and Applied Math., 54, Marcel Dekker, New York, 1980

[24] Demuner D.P., Federson M., Gutierrez C., The Poincaré-Bendixson theorem on the Klein bottle for continuous vector fields, Discrete Contin. Dyn. Syst., 2009, 25(2), 495–509 http://dx.doi.org/10.3934/dcds.2009.25.495[Crossref] | Zbl 1191.37013

[25] Denjoy A., Sur les courbes définies par les équations differentielles à la surface du tore, J. Math. Pures Appl., 1932, 11, 333–375 | Zbl 58.1124.04

[26] Erle D., Stable closed orbits in plane autonomous dynamical systems, J. Reine Angew. Math., 1979, 305, 136–139 | Zbl 0395.34054

[27] Fiedler B., Mallet-Paret J., A Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rational Mech. Anal., 1989, 107(4), 325–345 http://dx.doi.org/10.1007/BF00251553[Crossref] | Zbl 0704.35070

[28] Filippov V.V., Topological structure of solution spaces of ordinary differential equations, Russian Math. Surveys, 1993, 48(1), 101–154 http://dx.doi.org/10.1070/RM1993v048n01ABEH000986[Crossref] | Zbl 0808.34002

[29] Gutiérrez C., Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory Dynam. Systems, 1986, 6(1), 17–44

[30] Haas F., Poincaré-Bendixson type theorems for two-dimensional manifolds different from the torus, Ann. of Math., 1954, 59(2), 292–299 http://dx.doi.org/10.2307/1969694[Crossref] | Zbl 0057.06902

[31] Hájek O., Sections of dynamical systems in E 2, Czechoslovak Math. J., 1965, 15(90), 205–211 | Zbl 0134.42304

[32] Hájek O., Structure of dynamical systems, Comment. Math. Univ. Carolinae, 1965, 6, 53–72 | Zbl 0134.42302

[33] Hájek O., Dynamical Systems in the Plane, Academic Press, London-New York, 1968 | Zbl 0169.54401

[34] Hartman P., Ordinary Differential Equations, John Wiley & Sons, New York-London-Sydney, 1964

[35] Hastings H.M., A higher dimensional Poincaré-Bendixson theorem, Glas. Mat. Ser. III, 1979, 14(34)(2), 263–268 | Zbl 0435.34020

[36] Hirsch M.W., Pugh C.C., Cohomology of chain recurrent sets, Ergodic Theory Dynam. Systems, 1988, 8(1), 73–80 http://dx.doi.org/10.1017/S0143385700004326[Crossref] | Zbl 0643.54039

[37] Izydorek M., Rybicki S., Szafraniec Z., A note of the Poincaré-Bendixson index theorem, Kodai Math. J, 1996, 19(2), 145–156 http://dx.doi.org/10.2996/kmj/1138043594[Crossref] | Zbl 0863.34045

[38] Jiang J., Liang X., Competitive systems with migration and the Poincaré-Bendixson theorem for a 4-dimensional case, Quart. Appl. Math., 2006, 64(3), 483–498 [Crossref] | Zbl 1121.34053

[39] Jiménez López V., Soler López G., A topological characterization of !-limit sets for continuous flows on the projective plane, In: Dynamical Systems and Differential Equations, Kennesaw, 2000, Discrete Contin. Dynam. Systems, 2001, Added Volume, 254–258 | Zbl 1301.37028

[40] Jiménez López V., Soler López G., A characterization of ω-limit sets for continuous flows on surfaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 2006, 9(2), 515–521 | Zbl 1178.37015

[41] Jones D.S., Plank M.J., Sleeman B.D., Differential Equations and Mathematical Biology, 2nd ed., Chapman Hall/CRC Math. Comput. Biol. Ser., CRC Press, Boca Raton, 2010

[42] Klebanov B.S., On asymptotically autonomous differential equations in the plane, Topol. Methods Nonlinear Anal., 1997, 10(2), 327–338 | Zbl 0914.34051

[43] Kneser H., Reguläre Kurvenscharen auf Ringflächen, Math. Ann., 1924, 91(1–2), 135–154 http://dx.doi.org/10.1007/BF01498385[Crossref]

[44] Kuperberg K., A smooth counterexample to the Seifert conjecture, Ann. of Math., 1994, 140(3), 723–732 http://dx.doi.org/10.2307/2118623[Crossref] | Zbl 0856.57024

[45] Logan J.D., Wolesensky W.R., Mathematical Methods in Biology, Pure Appl. Math. (Hoboken), John Wiley & Sons, Hoboken, 2009

[46] Markley N.G., The Poincaré-Bendixson theorem for the Klein bottle, Trans. Amer. Math. Soc., 1969, 135, 159–165 | Zbl 0175.50101

[47] Markley N.G., On the number of recurrent orbit closures, Proc. Amer. Math. Soc., 1970, 25, 413–416 http://dx.doi.org/10.1090/S0002-9939-1970-0256375-0[Crossref] | Zbl 0198.56901

[48] Markoff A., Sur une propriété générale des ensembles minimaux de M. Birkhoff, C. R. Acad. Sci. Paris, 1931, 193, 823–825 | Zbl 57.1523.02

[49] Markus L., Asymptotically autonomous differential systems, In: Contributions to the Theory of Nonlinear Oscillations, 3, Ann. of Math. Stud., 36, Princeton University Press, Princeton, 1956, 17–29 | Zbl 0075.27002

[50] Mallet-Paret J., Sell G.R., The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 1996, 125(2), 441–489 http://dx.doi.org/10.1006/jdeq.1996.0037[Crossref] | Zbl 0849.34056

[51] McCann R.C., Negative escape time in semidynamical systems, Funkcial. Ekvac., 1977, 20(1), 39–47 | Zbl 0363.34034

[52] Melin J., Does distribution theory contain means for extending Poincaré-Bendixson theory? J. Math. Anal. Appl., 2005, 303(1), 81–89 http://dx.doi.org/10.1016/j.jmaa.2004.06.069[Crossref] | Zbl 1073.34007

[53] Neumann D.A., Existence of periodic orbits on 2-manifolds, J. Differential Equations, 1978, 27(3), 313–319 http://dx.doi.org/10.1016/0022-0396(78)90056-6

[54] Neumann D., Smoothing continuous flows on 2-manifolds, J. Differential Equations, 1978, 28(3), 327–344 http://dx.doi.org/10.1016/0022-0396(78)90131-6

[55] Opial Z., Sur la dépendance des solutions d’un systéme d’equations différentielles de leurs seconds membres. Application aux systémes presque autonomes, Ann. Polon. Math., 1960, 8, 75–89 | Zbl 0093.09002

[56] Peixoto M.M., Structural stability on two-dimensional manifolds, Topology, 1962, 1, 101–120 http://dx.doi.org/10.1016/0040-9383(65)90018-2[Crossref]

[57] Perko L., Differential Equations and Dynamical Systems, 3rd ed., Texts Appl. Math., 7, Springer, New York, 2001

[58] Poincaré H., Mémoire sur les courbes définies par une équation différentielle I, J. Math. Pures Appl. (3), 1881, 7, 375–422 | Zbl 13.0591.01

[59] Poincaré H., Mémoire sur les courbes définies par une équation différentielle II, J. Math. Pures Appl. (3), 1882, 8, 251–296 | Zbl 14.0666.01

[60] Poincaré H., Mémoire sur les courbes définies par une équation différentielle III, J. Math. Pures Appl. (4), 1885, 1, 167–244

[61] Poincaré H., Mémoire sur les courbes définies par une équation différentielle IV, J. Math. Pures Appl. (4), 1886, 2, 151–217

[62] Pokrovskii A.V, Pokrovskiy A.A., Zhezherun A., A corollary of the Poincaré-Bendixson theorem and periodic canards, J. Differential Equations, 2009, 247(12), 3283–3294 http://dx.doi.org/10.1016/j.jde.2009.09.010[Crossref] | Zbl 1229.34088

[63] Sacker R.J., Sell G.R., On the existence of periodic solutions on 2-manifolds, J. Differential Equations, 1972, 11, 449–463 http://dx.doi.org/10.1016/0022-0396(72)90058-7[Crossref]

[64] Saito T., Lectures on the Local Theory of Dynamical Systems, University of Minnesota, Minneapolis, 1960

[65] Sanchez L.A., Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems, J. Differential Equations, 2009, 246(5), 1978–1990 http://dx.doi.org/10.1016/j.jde.2008.10.015[Crossref]

[66] Schwartz A.J., A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds, Amer. J. Math., 1963, 85, 453–458 http://dx.doi.org/10.2307/2373135[Crossref] | Zbl 0116.06803

[67] Schweitzer P.A., Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math., 1974, 100, 386–400 http://dx.doi.org/10.2307/1971077[Crossref] | Zbl 0295.57010

[68] Seibert P., Tulley P., On dynamical systems in the plane, Arch. Math. (Basel), 1967, 18, 290–292 http://dx.doi.org/10.1007/BF01900636[Crossref] | Zbl 0152.40301

[69] Seifert H., Closed integral curves in 3-space and isotopic two-dimensional deformations, Proc. Amer. Math. Soc., 1950, 1, 287–302 | Zbl 0039.40002

[70] Serra E., Tarallo M., A new proof of the Poincaré-Bendixson theorem, Riv. Mat. Pura Appl., 1990, 7, 81–86 | Zbl 0723.34031

[71] Smith R.A., Existence of periodic orbits of autonomous ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 1980, 85(1–2), 153–172 http://dx.doi.org/10.1017/S030821050001177X[Crossref] | Zbl 0429.34040

[72] Smith R.A., Orbital stability for ordinary differential equations, J. Differential Equations, 1987, 69(2), 265–287 http://dx.doi.org/10.1016/0022-0396(87)90120-3[Crossref]

[73] Smith R.A., Poincaré-Bendixson theory for certain retarded functional-differential equations, Differential Integral Equations, 1992, 5(1), 213–240 | Zbl 0754.34070

[74] Solntzev G., On the asymptotic behaviour of integral curves of a system of differential equations, Bull. Acad. Sci. URSS. Sér. Math., 1945, 9, 233–240 (in Russian) | Zbl 0061.19702

[75] Thieme H.R., Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 1992, 30(7), 755–763 http://dx.doi.org/10.1007/BF00173267[Crossref] | Zbl 0761.34039

[76] Tu P.N.V., Dynamical Systems, 2nd ed., Springer, Berlin, 1994 http://dx.doi.org/10.1007/978-3-642-78793-5

[77] Ura T., Kimura I., Sur le courant extérieur à une région invariante; théorème de Bendixson, Comment. Math. Univ. St. Paul, 1960, 8, 23–39 | Zbl 0117.16103

[78] Verhulst F., Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, Berlin, 1990 http://dx.doi.org/10.1007/978-3-642-97149-5[Crossref]

[79] Vinograd R., On the limiting behavior of unbounded integral curves, Doklady Akad. Nauk SSSR (N.S.), 1949, 66, 5–8 (in Russian)

[80] Vinograd R.È., On the limit behavior of an unbounded integral curve, Moskov. Gos. Univ. Uč. Zap. Mat., 1952, 155,5, 94–136 (in Russian)

[81] Whitney H., Regular families of curves I–II, Proc. Natl. Acad. Sci. USA, 1932, 18(3,4), 275–278, 340–342 http://dx.doi.org/10.1073/pnas.18.3.275[Crossref] | Zbl 0004.07503

[82] Whitney H., Regular families of curves, Ann. of Math., 1933, 34(2), 244–270 http://dx.doi.org/10.2307/1968202[Crossref] | Zbl 0006.37101