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A constructive proof of the Poincaré-Birkhoff theorem


Authors: Yong Li and Zheng Hua Lin
Journal: Trans. Amer. Math. Soc. 347 (1995), 2111-2126
MSC: Primary 58F22; Secondary 34C25
DOI: https://doi.org/10.1090/S0002-9947-1995-1290734-4
MathSciNet review: 1290734
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Abstract: In this paper, with the use of the homotopy method, a constructive proof of the Poincaré-Birkhoff theorem is given. This approach provides a global method for finding fixed points of area-preserving maps and periodic solutions of Duffing equations.


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  • [1] J. C. Alexander and J. A. Yorke, The homotopy continuation method: Numerically implementable topological procedures, Trans. Amer. Math. Soc. 242 (1978), 271-284. MR 0478138 (57:17627)
  • [2] E. L. Allgower and K. Georg, Simplicial continuation methods for approximating fixed points and solutions to systems of equations, SIAM Rev. 22 (1980), 28-85. MR 554709 (81d:47040)
  • [3] -, Numerical continuation methods: An introduction, Springer-Verlag, Berlin, New York, 1990. MR 1059455 (92a:65165)
  • [4] S. Alpern and V. S. Prasad, Combinatorial proofs of the Conley-Zehnder-Franks theorem on a fixed point for torus homeomorphisms, Adv. in Math. 99 (1993), 238-247. MR 1219584 (94c:58101)
  • [5] -, Fixed points of area-preserving annulus homeomorphisms, Fixed Point Theory and Applications (M.A. Thera and J.B. Baillon, Eds.), Pitman Research Notes in Math., 252, 1991, pp. 1-8. MR 1122814 (92h:58158)
  • [6] V. I. Arnold, Fixed points of symplectic diffeomorphisms, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R.I., 1976.
  • [7] G. D. Birkhoff, Proof of Poincare's last geometric theorem, Trans. Amer. Math. Soc. 14 (1913), 333-343.
  • [8] -, An extension of Poincare's last geometric theorem, Acta Math. 47 (1925), 297-311.
  • [9] -, Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 27, Amer. Math. Soc. Providence, R.I., 1927; revised 1966; reprinted 1979.
  • [10] -, Une generalisation $ a$-$ n$-dimensions du dernier theoreme de geometrie de Poincaré, C.R. Acad. Sci. 192 (1931), 196-198.
  • [11] S-N. Chow, J. Mallet-Paret and J. A. Yorke, Finding zeros of maps: Homotopy methods that are constructive with probability one, Math. Comp. 143 (1978), 887-899. MR 492046 (80d:55002)
  • [12] C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold, Invent. Math. 73 (1983), 33-49. MR 707347 (85e:58044)
  • [13] C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207-253. MR 733717 (86b:58021)
  • [14] M. A. Del Pino, R. Manaserich and A. E. Murna, On the number of $ 2\pi $-periodic solutions for $ u'' + g(u) = s(1 + h(t))$ using the Poincare-Birkhoff theorem, J. Differential Equations 95 (1992), 240-258.
  • [15] R. Dieckerhoff and E. Zehnder, An "a-priori" estimate for oscillatory- equations, Lecture Notes in Math., vol. 1125, Springer, 1985. MR 798079
  • [16] T. Ding, Nonlinear oscillations at point of resonance, Sci. Sinica Ser. A 25 ( 1982), 918-931. MR 681856 (84c:34058)
  • [17] -, An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance, Proc. Amer. Math. Soc. 86 (1982), 47-54. MR 663864 (83j:34041)
  • [18] T. Ding and W. Ding, Resonance problem for a class of Duffing equations, Chinese Ann. Math. Ser. B 6 (1985), 427-432. MR 843678 (87i:34041)
  • [19] T. Ding and F. Zanolin, Time-maps for the solvability of periodically perturbed nonlinear Duffing equations, Nonlinear Analysis, TMA, 1991. MR 1128965 (93j:34053)
  • [20] W. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations, Acta Math. Sinica (Chinese) 25 (1982), 227-235. MR 677834 (84d:58061)
  • [21] -, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc. 88 (1983), 341-346. MR 695272 (84f:54053)
  • [22] M. Flucher, Fixed points of measure preserving torus homeomorphisms, Manuscripta Math. 68 (1990), 271-293. MR 1065931 (91j:58129)
  • [23] A. Fonda, R. Manaserich and F. Zanolin, Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal. 24 (1993), 1294-1311. MR 1234017 (94f:34085)
  • [24] J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynamical Systems 8 (1988), 99-107. MR 967632 (90d:58124)
  • [25] J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math. 128 (1988), 139-151. MR 951509 (89m:54052)
  • [26] C. B. Garcia and W. I. Zangwill, An approach to homotopy and degree theory, Math. Oper. Res. 4 (1979), 390-405. MR 549125 (81d:55004)
  • [27] -, Pathways to solution, fixed points, and equilibria, Prentice-Hall, Englewood Cliffs, NJ, 1981.
  • [28] P. Hartman, On boundary value problems for superlinear second order differential equations, J. Differential Equations 26 (1977), 37-53. MR 0477242 (57:16783)
  • [29] H. Jacobowitz, Periodic solutions of $ x'' + f(x,t) = 0$ via the Poincaré-Birkhoff theorem (and Corrigendum: The existence of the second fixed point), J. Differential Equations 20 (1976), 37-52; 25 (1977),148-149. MR 0393673 (52:14482)
  • [30] H. B. Keller, Global homotopies and Newton methods, Recent Advances in Numerical Analysis (C. de Boor and G. Golub, Eds.), Academic Press, New York., 1978, pp. 73-94. MR 519057 (80f:65059)
  • [31] R. B. Kellogg, T. Y. Li and J. A. Yorke, A constructive proof of the Brouwer fixed point theorem and computational results, SIAM J. Numer. Anal. 13 (1976), 473-483. MR 0416010 (54:4087)
  • [32] M. Levi, KAM theory for particles in periodic potentials, Ergodic Theory Dynamical Systems 10 (1990), 777-785. MR 1091426 (92a:58118)
  • [33] Li Yong and Lu Xianrui, Continuation theorems to boundary value problems, J. Math. Anal. Appl. 188 (1994). MR 1314105 (95m:34039)
  • [34] B. Liu, Boundedness for solutions of nonlinear Hill's equations with periodic forcing terms via Moser's twist theorem, J. Differential Equations 79 (1989), 304-315. MR 1000692 (90k:34050)
  • [35] -, An existence theorem for harmonic solutions of semi-linear Duffing equations, Acta Math. Sinica 34 (1991), 165-170. MR 1117074 (92j:34089)
  • [36] G. Morris, A case of boundedness in Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc. 14 (1976), 71-93. MR 0402198 (53:6019)
  • [37] J. Moser, On invariant curves of area-preserving mapping of annulus, Nachr. Akad. Wiss. Gottingen Math. Phys. K. 2 (1962), 1-20. MR 0147741 (26:5255)
  • [38] H. Poincaré, Sur un theoreme de geometrie, Rend. Circ. Mat. Palermo 33 (1912), 375-407.
  • [39] S. Smale, A convergent process of price adjustment and global Newton methods, J. Math. Econom. 3 (1976), 1-14. MR 0411577 (53:15310)
  • [40] L. T. Watson and M. R. Scott, Solving spline-collocation approximations to nonlinear two-point boundary value problems by a homotopy method, Appl. Math. Comput. 24 (1987), 333-357. MR 919208 (89a:65122)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1290734-4
Keywords: Constructive proof, the Poincaré-Birkhoff theorem, periodic solutions of Duffing equations
Article copyright: © Copyright 1995 American Mathematical Society

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