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

Krzysztof Ciesielski

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 2110-2128
  • ISSN: 2391-5455

Abstract

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

How to cite

top

Krzysztof 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. [1] Aarts J.M., de Vries J., Colloquium Topologische Dynamische Systemen, MC Syllabus, 36, Mathematisch Centrum, Amsterdam, 1977 
  2. [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. [3] Arrowsmith D.K., Place C.M., Ordinary Differential Equations, Chapman and Hall Math. Ser., Chapman & Hall, London, 1982 Zbl0481.34005
  4. [4] Athanassopoulos K., Flows with cyclic winding number groups, J. Reine Angew. Math., 1996, 481, 207–215 Zbl0858.58041
  5. [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. [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. [7] Athanassopoulos K., Strantzalos P., On minimal sets in 2-manifolds, J. Reine Angew. Math., 1988, 388, 206–211 Zbl0647.58027
  8. [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. [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. [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. [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. [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. [13] Bhatia N.P., Hájek O., Local Semi-Dynamical Systems, Lecture Notes in Math., 90, Springer, Berlin-New York, 1969 Zbl0176.39102
  14. [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. [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. [16] Birkhoff G.D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., 9, American Mathematical Society, Providence, 1927 Zbl53.0732.01
  17. [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. [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. [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. [20] Ciesielski K., Sections in semidynamical systems, Bull. Polish Acad. Sci. Math., 1992, 40, 61–70 Zbl0602.54040
  21. [21] Ciesielski K., The Poincaré-Bendixson theorems for two-dimensional semiflows, Topol. Methods Nonlinear Anal., 1994, 3(1), 163–178 Zbl0839.54028
  22. [22] Coddington E.A., Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill, New York-Toronto-London, 1955 Zbl0064.33002
  23. [23] Cronin J., Differential Equations, Monographs and Textbooks in Pure and Applied Math., 54, Marcel Dekker, New York, 1980 
  24. [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. [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. [26] Erle D., Stable closed orbits in plane autonomous dynamical systems, J. Reine Angew. Math., 1979, 305, 136–139 Zbl0395.34054
  27. [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. [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. [29] Gutiérrez C., Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory Dynam. Systems, 1986, 6(1), 17–44 
  30. [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. [31] Hájek O., Sections of dynamical systems in E 2, Czechoslovak Math. J., 1965, 15(90), 205–211 Zbl0134.42304
  32. [32] Hájek O., Structure of dynamical systems, Comment. Math. Univ. Carolinae, 1965, 6, 53–72 Zbl0134.42302
  33. [33] Hájek O., Dynamical Systems in the Plane, Academic Press, London-New York, 1968 Zbl0169.54401
  34. [34] Hartman P., Ordinary Differential Equations, John Wiley & Sons, New York-London-Sydney, 1964 
  35. [35] Hastings H.M., A higher dimensional Poincaré-Bendixson theorem, Glas. Mat. Ser. III, 1979, 14(34)(2), 263–268 Zbl0435.34020
  36. [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. [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. [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. [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. [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. [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. [42] Klebanov B.S., On asymptotically autonomous differential equations in the plane, Topol. Methods Nonlinear Anal., 1997, 10(2), 327–338 Zbl0914.34051
  43. [43] Kneser H., Reguläre Kurvenscharen auf Ringflächen, Math. Ann., 1924, 91(1–2), 135–154 http://dx.doi.org/10.1007/BF01498385[Crossref] 
  44. [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. [45] Logan J.D., Wolesensky W.R., Mathematical Methods in Biology, Pure Appl. Math. (Hoboken), John Wiley & Sons, Hoboken, 2009 
  46. [46] Markley N.G., The Poincaré-Bendixson theorem for the Klein bottle, Trans. Amer. Math. Soc., 1969, 135, 159–165 Zbl0175.50101
  47. [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. [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. [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. [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. [51] McCann R.C., Negative escape time in semidynamical systems, Funkcial. Ekvac., 1977, 20(1), 39–47 Zbl0363.34034
  52. [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. [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. [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. [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. [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. [57] Perko L., Differential Equations and Dynamical Systems, 3rd ed., Texts Appl. Math., 7, Springer, New York, 2001 
  58. [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. [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. [60] Poincaré H., Mémoire sur les courbes définies par une équation différentielle III, J. Math. Pures Appl. (4), 1885, 1, 167–244 
  61. [61] Poincaré H., Mémoire sur les courbes définies par une équation différentielle IV, J. Math. Pures Appl. (4), 1886, 2, 151–217 
  62. [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. [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. [64] Saito T., Lectures on the Local Theory of Dynamical Systems, University of Minnesota, Minneapolis, 1960 
  65. [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. [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. [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. [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. [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. [70] Serra E., Tarallo M., A new proof of the Poincaré-Bendixson theorem, Riv. Mat. Pura Appl., 1990, 7, 81–86 Zbl0723.34031
  71. [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. [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. [73] Smith R.A., Poincaré-Bendixson theory for certain retarded functional-differential equations, Differential Integral Equations, 1992, 5(1), 213–240 Zbl0754.34070
  74. [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. [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. [76] Tu P.N.V., Dynamical Systems, 2nd ed., Springer, Berlin, 1994 http://dx.doi.org/10.1007/978-3-642-78793-5 
  77. [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. [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. [79] Vinograd R., On the limiting behavior of unbounded integral curves, Doklady Akad. Nauk SSSR (N.S.), 1949, 66, 5–8 (in Russian) 
  80. [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. [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. [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 ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.