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International Journal of Differential Equations
Volume 2011 (2011), Article ID 612041, 20 pages
doi:10.1155/2011/612041
Research Article
A Topological Approach to Bend-Twist Maps with Applications
Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, 33100 Udine, Italy
Received 26 May 2011; Accepted 21 July 2011
Academic Editor: Leonid Berezansky
Copyright © 2011 Anna Pascoletti and Fabio Zanolin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- H. Poincaré, “Sur un théorème de geométrie,” Rendiconti del Circolo Matematico di Palermo, vol. 33, pp. 375–389, 1912. View at Publisher · View at Google Scholar
- G. D. Birkhoff, “Proof of Poincaré's geometric theorem,” Transactions of the American Mathematical Society, vol. 14, pp. 14–22, 1913.
- G. D. Birkhoff, “An extension of Poincaré's last geometric theorem,” Acta Mathematica, vol. 47, no. 4, pp. 297–311, 1925. View at Publisher · View at Google Scholar · View at MathSciNet
- M. Brown and W. D. Neumann, “Proof of the Poincaré-Birkhoff fixed point theorem,” Michigan Mathematical Journal, vol. 24, no. 1, pp. 21–31, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- F. Dalbono and C. Rebelo, “Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems,” Rendiconti del Seminario Matematico Università e Politecnico di Torino, vol. 60, no. 4, pp. 233–263, 2003.
- R. Martins and A. J. Ureña, “The star-shaped condition on Ding's version of the Poincaré-Birkhoff theorem,” Bulletin of the London Mathematical Society, vol. 39, no. 5, pp. 803–810, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- P. Le Calvez and J. Wang, “Some remarks on the Poincaré-Birkhoff theorem,” Proceedings of the American Mathematical Society, vol. 138, no. 2, pp. 703–715, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- T. Ding, Approaches to the Qualitative Theory of Ordinary Differential Equations, vol. 3 of Peking University Series in Mathematics, World Scientific, Hackensack, NJ, USA, 2007.
- A. Pascoletti and F. Zanolin, “A crossing lemma for annular regions and invariant sets,” Preprint. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- E. E. Moise, Geometric Topology in Dimensions 2 and 3, Springer, New York, NY, USA, 1977.
- A. Berarducci, D. Dikranjan, and J. Pelant, “Uniform quasi components, thin spaces and compact separation,” in Proceedings of the International Conference on Topology and its Applications (Yokohama, 1999), vol. 122, pp. 51–64, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. G. Hocking and G. S. Young, Topology, Dover Publications, New York, NY, USA, 2nd edition, 1988.
- D. E. Sanderson, “Advanced plane topology from an elementary standpoint,” Mathematics Magazine, vol. 53, no. 2, pp. 81–89, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- C. Rebelo, “A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems,” Nonlinear Analysis, vol. 29, no. 3, pp. 291–311, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- M. J. Greenberg, Lectures on Algebraic Topology, W. A. Benjamin Inc., New York, NY, USA, 1967.
- G. D. Birkhoff, “Sur la demonstration directe du renier théorème de Henri Poincaré par M. Dantzig,” Bulletin des Sciences Mathématiques, vol. 42, pp. 41–43, 1918.
- G. R. Morris, “An infinite class of periodic solutions of ,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 61, pp. 157–164, 1965. View at Zentralblatt MATH
- A. Capietto, J. Mawhin, and F. Zanolin, “Periodic solutions of some superlinear functional differential equations,” in Proceedings of the International Symposium on Functional Differential Equations (Kyoto, 1990), pp. 19–31, Singapore, 1991, World Scientific.
- M. Henrard and F. Zanolin, “Bifurcation from a periodic orbit in perturbed planar Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 277, no. 1, pp. 79–103, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. Margheri, C. Rebelo, and F. Zanolin, “Chaos in periodically perturbed planar Hamiltonian systems using linked twist maps,” Journal of Differential Equations, vol. 249, no. 12, pp. 3233–3257, 2010. View at Publisher · View at Google Scholar
- J. K. Hale, Ordinary Differential Equations, Robert E. Krieger, Huntington, NY, USA, 2nd edition, 1980.
- P. Buttazzoni and A. Fonda, “Periodic perturbations of scalar second order differential equations,” Discrete and Continuous Dynamical Systems, vol. 3, no. 3, pp. 451–455, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- W. Y. Ding, “A generalization of the Poincaré-Birkhoff theorem,” Proceedings of the American Mathematical Society, vol. 88, no. 2, pp. 341–346, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- F. Zanolin, “Time-maps and boundary value problems for ordinary differential equations,” in Tricomi's Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), vol. 147 of Atti Convegni Lincei, pp. 281–302, Accad. Naz. Lincei, Rome, Italy, 1998. View at Zentralblatt MATH
- W. D. Neumann, “Generalizations of the Poincaré Birkhoff fixed point theorem,” Bulletin of the Australian Mathematical Society, vol. 17, no. 3, pp. 375–389, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- J. Franks, “Generalizations of the Poincaré-Birkhoff theorem,” Annals of Mathematics. Second Series, vol. 128, no. 1, pp. 139–151, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Franks, “Erratum: generalizations of the Poincaré-Birkhoff theorem,” Annals of Mathematics. Second Series, vol. 164, no. 2, pp. 1097–1098, 2006.
- M. Henrard, “Degree for superlinear Floquet boundary value problems and existence of four solutions with a prescribed number of zeros,” Report 25/97/M, SISSA, 1997.