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

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Central European Journal of Mathematics

Abstract

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

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References

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

    MATH  Google Scholar 

  2. Abate M., Tovena F., Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields, J. Differential Equations, 2011, 251(9), 2612–2684

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Athanassopoulos K., One-dimensional chain recurrent sets of flows in the 2-sphere, Math. Z., 1996, 223(4), 643–649

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  Google Scholar 

  10. Beck A., Continuous Flows in the Plane, Grundlehren Math. Wiss., 201, Springer, New York-Heidelberg, 1974

    Book  MATH  Google Scholar 

  11. Bendixson I., Sur les courbes définies par des équations différentielles, Acta Math., 1901, 24(1), 1–88

    Article  MathSciNet  Google Scholar 

  12. Bhatia N.P., Attraction and nonsaddle sets in dynamical systems, J. Differential Equations, 1970, 8, 229–249

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. Bhatia N.P., Lazer A.C., Leighton W., Application of the Poincaré-Bendixson Theorem, Ann. Mat. Pura Appl., 1966, 73, 27–32

    Article  MathSciNet  MATH  Google Scholar 

  15. Bhatia N.P., Szegő G.P., Stability Theory of Dynamical Systems, Grundlehren Math. Wiss., 161, Springer, New York-Berlin, 1970

    Book  MATH  Google Scholar 

  16. Birkhoff G.D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., 9, American Mathematical Society, Providence, 1927

    MATH  Google Scholar 

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

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  20. Ciesielski K., Sections in semidynamical systems, Bull. Polish Acad. Sci. Math., 1992, 40, 61–70

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. Fiedler B., Mallet-Paret J., A Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rational Mech. Anal., 1989, 107(4), 325–345

    Article  MathSciNet  MATH  Google Scholar 

  28. Filippov V.V., Topological structure of solution spaces of ordinary differential equations, Russian Math. Surveys, 1993, 48(1), 101–154

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Haas F., Poincaré-Bendixson type theorems for two-dimensional manifolds different from the torus, Ann. of Math., 1954, 59(2), 292–299

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  32. Hájek O., Structure of dynamical systems, Comment. Math. Univ. Carolinae, 1965, 6, 53–72

    MathSciNet  MATH  Google Scholar 

  33. Hájek O., Dynamical Systems in the Plane, Academic Press, London-New York, 1968

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  36. Hirsch M.W., Pugh C.C., Cohomology of chain recurrent sets, Ergodic Theory Dynam. Systems, 1988, 8(1), 73–80

    Article  MathSciNet  MATH  Google Scholar 

  37. Izydorek M., Rybicki S., Szafraniec Z., A note of the Poincaré-Bendixson index theorem, Kodai Math. J, 1996, 19(2), 145–156

    Article  MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  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

  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

    MathSciNet  MATH  Google Scholar 

  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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  43. Kneser H., Reguläre Kurvenscharen auf Ringflächen, Math. Ann., 1924, 91(1–2), 135–154

    Article  MathSciNet  MATH  Google Scholar 

  44. Kuperberg K., A smooth counterexample to the Seifert conjecture, Ann. of Math., 1994, 140(3), 723–732

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Markley N.G., On the number of recurrent orbit closures, Proc. Amer. Math. Soc., 1970, 25, 413–416

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  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

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  52. Melin J., Does distribution theory contain means for extending Poincaré-Bendixson theory? J. Math. Anal. Appl., 2005, 303(1), 81–89

    Article  MathSciNet  MATH  Google Scholar 

  53. Neumann D.A., Existence of periodic orbits on 2-manifolds, J. Differential Equations, 1978, 27(3), 313–319

    Article  MathSciNet  MATH  Google Scholar 

  54. Neumann D., Smoothing continuous flows on 2-manifolds, J. Differential Equations, 1978, 28(3), 327–344

    Article  MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  56. Peixoto M.M., Structural stability on two-dimensional manifolds, Topology, 1962, 1, 101–120

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  63. Sacker R.J., Sell G.R., On the existence of periodic solutions on 2-manifolds, J. Differential Equations, 1972, 11, 449–463

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  66. Schwartz A.J., A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds, Amer. J. Math., 1963, 85, 453–458

    Article  MATH  Google Scholar 

  67. Schweitzer P.A., Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math., 1974, 100, 386–400

    Article  MathSciNet  MATH  Google Scholar 

  68. Seibert P., Tulley P., On dynamical systems in the plane, Arch. Math. (Basel), 1967, 18, 290–292

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  72. Smith R.A., Orbital stability for ordinary differential equations, J. Differential Equations, 1987, 69(2), 265–287

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  75. Thieme H.R., Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 1992, 30(7), 755–763

    Article  MathSciNet  MATH  Google Scholar 

  76. Tu P.N.V., Dynamical Systems, 2nd ed., Springer, Berlin, 1994

    Book  MATH  Google Scholar 

  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

    MathSciNet  Google Scholar 

  78. Verhulst F., Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, Berlin, 1990

    Book  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  81. Whitney H., Regular families of curves I–II, Proc. Natl. Acad. Sci. USA, 1932, 18(3,4), 275–278, 340–342

    Article  MATH  Google Scholar 

  82. Whitney H., Regular families of curves, Ann. of Math., 1933, 34(2), 244–270

    Article  MathSciNet  Google Scholar 

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  1. Mathematics Institute, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-364, Kraków, Poland

    Krzysztof Ciesielski

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Ciesielski, K. The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century. centr.eur.j.math. 10, 2110–2128 (2012). https://doi.org/10.2478/s11533-012-0110-y

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