The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century
Open Mathematics (2012)
- Volume: 10, Issue: 6, page 2110-2128
- ISSN: 2391-5455
Access Full Article
topAbstract
topHow to cite
topKrzysztof Ciesielski. "The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century." Open Mathematics 10.6 (2012): 2110-2128. <http://eudml.org/doc/269766>.
@article{KrzysztofCiesielski2012,
abstract = {The Poincaré-Bendixson Theorem and the development of the theory are presented - from the papers of Poincaré and Bendixson to modern results.},
author = {Krzysztof Ciesielski},
journal = {Open Mathematics},
keywords = {Poincaré-Bendixson Theorem; Limit set; Flow; 2-dimensional system; Periodic trajectory; Critical point; Section; Poincaré-Bendixson theorem; limit set; flow; periodic trajectory; critical point; section},
language = {eng},
number = {6},
pages = {2110-2128},
title = {The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century},
url = {http://eudml.org/doc/269766},
volume = {10},
year = {2012},
}
TY - JOUR
AU - Krzysztof Ciesielski
TI - The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 2110
EP - 2128
AB - The Poincaré-Bendixson Theorem and the development of the theory are presented - from the papers of Poincaré and Bendixson to modern results.
LA - eng
KW - Poincaré-Bendixson Theorem; Limit set; Flow; 2-dimensional system; Periodic trajectory; Critical point; Section; Poincaré-Bendixson theorem; limit set; flow; periodic trajectory; critical point; section
UR - http://eudml.org/doc/269766
ER -
References
top- [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] Zbl1241.32012
- [3] Arrowsmith D.K., Place C.M., Ordinary Differential Equations, Chapman and Hall Math. Ser., Chapman & Hall, London, 1982 Zbl0481.34005
- [4] Athanassopoulos K., Flows with cyclic winding number groups, J. Reine Angew. Math., 1996, 481, 207–215 Zbl0858.58041
- [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] Zbl0872.58051
- [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] Zbl0927.37023
- [7] Athanassopoulos K., Strantzalos P., On minimal sets in 2-manifolds, J. Reine Angew. Math., 1988, 388, 206–211 Zbl0647.58027
- [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] Zbl0295.54001
- [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] Zbl31.0328.03
- [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] Zbl0207.08803
- [13] Bhatia N.P., Hájek O., Local Semi-Dynamical Systems, Lecture Notes in Math., 90, Springer, Berlin-New York, 1969 Zbl0176.39102
- [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] Zbl0144.22403
- [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] Zbl0213.10904
- [16] Birkhoff G.D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., 9, American Mathematical Society, Providence, 1927 Zbl53.0732.01
- [17] Bohr H., Fenchel W., Ein Satz über stabile Bewegungen in der Ebene, Danske Vid. Selsk. Mat.-Fys. Medd., 1936, 14(1), 1–15 Zbl62.0919.02
- [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] Zbl1143.37014
- [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 Zbl0273.54029
- [20] Ciesielski K., Sections in semidynamical systems, Bull. Polish Acad. Sci. Math., 1992, 40, 61–70 Zbl0602.54040
- [21] Ciesielski K., The Poincaré-Bendixson theorems for two-dimensional semiflows, Topol. Methods Nonlinear Anal., 1994, 3(1), 163–178 Zbl0839.54028
- [22] Coddington E.A., Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill, New York-Toronto-London, 1955 Zbl0064.33002
- [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] Zbl1191.37013
- [25] Denjoy A., Sur les courbes définies par les équations differentielles à la surface du tore, J. Math. Pures Appl., 1932, 11, 333–375 Zbl58.1124.04
- [26] Erle D., Stable closed orbits in plane autonomous dynamical systems, J. Reine Angew. Math., 1979, 305, 136–139 Zbl0395.34054
- [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] Zbl0704.35070
- [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] Zbl0808.34002
- [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] Zbl0057.06902
- [31] Hájek O., Sections of dynamical systems in E 2, Czechoslovak Math. J., 1965, 15(90), 205–211 Zbl0134.42304
- [32] Hájek O., Structure of dynamical systems, Comment. Math. Univ. Carolinae, 1965, 6, 53–72 Zbl0134.42302
- [33] Hájek O., Dynamical Systems in the Plane, Academic Press, London-New York, 1968 Zbl0169.54401
- [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 Zbl0435.34020
- [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] Zbl0643.54039
- [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] Zbl0863.34045
- [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] Zbl1121.34053
- [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 Zbl1301.37028
- [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 Zbl1178.37015
- [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 Zbl0914.34051
- [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] Zbl0856.57024
- [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 Zbl0175.50101
- [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] Zbl0198.56901
- [48] Markoff A., Sur une propriété générale des ensembles minimaux de M. Birkhoff, C. R. Acad. Sci. Paris, 1931, 193, 823–825 Zbl57.1523.02
- [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 Zbl0075.27002
- [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] Zbl0849.34056
- [51] McCann R.C., Negative escape time in semidynamical systems, Funkcial. Ekvac., 1977, 20(1), 39–47 Zbl0363.34034
- [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] Zbl1073.34007
- [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 Zbl0093.09002
- [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 Zbl13.0591.01
- [59] Poincaré H., Mémoire sur les courbes définies par une équation différentielle II, J. Math. Pures Appl. (3), 1882, 8, 251–296 Zbl14.0666.01
- [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] Zbl1229.34088
- [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] Zbl0116.06803
- [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] Zbl0295.57010
- [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] Zbl0152.40301
- [69] Seifert H., Closed integral curves in 3-space and isotopic two-dimensional deformations, Proc. Amer. Math. Soc., 1950, 1, 287–302 Zbl0039.40002
- [70] Serra E., Tarallo M., A new proof of the Poincaré-Bendixson theorem, Riv. Mat. Pura Appl., 1990, 7, 81–86 Zbl0723.34031
- [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] Zbl0429.34040
- [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 Zbl0754.34070
- [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) Zbl0061.19702
- [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] Zbl0761.34039
- [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 Zbl0117.16103
- [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] Zbl0004.07503
- [82] Whitney H., Regular families of curves, Ann. of Math., 1933, 34(2), 244–270 http://dx.doi.org/10.2307/1968202[Crossref] Zbl0006.37101
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.