Table of Contents Author Guidelines Submit a Manuscript
Journal of Mathematics
Volume 2013, Article ID 267393, 12 pages
http://dx.doi.org/10.1155/2013/267393
Research Article

A Crossing Lemma for Annular Regions and Invariant Sets with an Application to Planar Dynamical Systems

Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy

Received 31 January 2013; Accepted 17 June 2013

Academic Editor: Alfredo Peris

Copyright © 2013 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

  1. H. Poincaré, “Sur certaines solutions particulières du problème des trois corps,” Comptes Rendus de l'Académie des Sciences, vol. 97, pp. 251–252, 1883. View at Google Scholar
  2. H. Poincaré, “Sur certaines solutions particulières du problème des trois corps,” Bulletin Astronomique, vol. 1, pp. 65–74, 1884. View at Google Scholar
  3. F. E. Browder, “Fixed point theory and nonlinear problems,” Bulletin of the American Mathematical Society, vol. 9, no. 1, pp. 1–39, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. W. Kulpa, “The Poincaré-Miranda theorem,” The American Mathematical Monthly, vol. 104, no. 6, pp. 545–550, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Mawhin, “Poincaré's early use of Analysis situs in nonlinear differential equations: variations around the theme of Kronecker’s integral,” Philosophia Scientiae, vol. 4, no. 1, pp. 103–143, 2000. View at Google Scholar
  6. H. W. Siegberg, “Some historical remarks concerning degree theory,” The American Mathematical Monthly, vol. 88, no. 2, pp. 125–139, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Mawhin, “Variations on Poincaré's-Miranda’s Theorem,” Advanced Nonlinear Studies, vol. 13, pp. 209–217, 2013. View at Google Scholar
  8. A. Pascoletti, M. Pireddu, and F. Zanolin, “Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps,” in The 8th Colloquium on the Qualitative Theory of Differential Equations, vol. 8, Szeged, 2008, Electronic Journal of Qualitative Theory of Differential Equations, Szeged, vol. 14, pp. 1–32, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. W. Hurewicz and H. Wallman, Dimension Theory, vol. 4 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1941. View at MathSciNet
  10. S. Alpern and A. Beck, “Hex games and twist maps on the annulus,” The American Mathematical Monthly, vol. 98, no. 9, pp. 803–811, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. D. Gale, “The game of Hex and the Brouwer fixed-point theorem,” The American Mathematical Monthly, vol. 86, no. 10, pp. 818–827, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. C. Conley, “An application of Ważewski's method to a non-linear boundary value problem which arises in population genetics,” Journal of Mathematical Biology, vol. 2, no. 3, pp. 241–249, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. C. Alexander, “A primer on connectivity,” in Fixed Point Theory, vol. 886 of Lecture Notes in Math, pp. 455–483, Springer, Berlin, Germany, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. G. J. Butler, “Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations,” Journal of Differential Equations, vol. 22, no. 2, pp. 467–477, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  15. C. Rebelo and F. Zanolin, “On the existence and multiplicity of branches of nodal solutions for a class of parameter-dependent Sturm-Liouville problems via the shooting map,” Differential and Integral Equations, vol. 13, no. 10–12, pp. 1473–1502, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Pascoletti and F. Zanolin, “A path crossing lemma and applications to nonlinear second order equations under slowly varying perturbations,” Le Matematiche, vol. 65, no. 2, pp. 121–168, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. A. Kennedy and J. A. Yorke, “The topology of stirred fluids,” Topology and Its Applications, vol. 80, no. 3, pp. 201–238, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. Berarducci, D. Dikranjan, and J. Pelant, “Uniform quasi components, thin spaces and compact separation,” Topology and Its Applications, vol. 122, pp. 51–64, 2002. View at Google Scholar
  19. M. Dolcher, “Questioni di minimo per insiemi chiusi sconnettenti uno spazio topologico,” The Mathematical Journal of the University of Padova, vol. 19, pp. 159–171, 1950. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. Papini and F. Zanolin, “Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells,” Fixed Point Theory and Applications, vol. 2004, no. 2, pp. 113–134, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  21. A. Pascoletti and F. Zanolin, “Example of a suspension bridge ODE model exhibiting chaotic dynamics: a topological approach,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1179–1198, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  22. M. Pireddu and F. Zanolin, “Cutting surfaces and applications to periodic points and chaotic-like dynamics,” Topological Methods in Nonlinear Analysis, vol. 30, no. 2, pp. 279–319, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. E. E. Moise, Geometric Topology in Dimensions 2 and 3, vol. 47 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1977. View at MathSciNet
  24. J. S. Muldowney and D. Willett, “An elementary proof of the existence of solutions to second order nonlinear boundary value problems,” SIAM Journal on Mathematical Analysis, vol. 5, pp. 701–707, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Dover, Mineola, NY, USA, 2003. View at MathSciNet
  26. A. Pascoletti and F. Zanolin, “Chaotic dynamics in periodically forced asymmetric ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 352, no. 2, pp. 890–906, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. W. Alexander, “A proof of Jordan's theorem about a simple closed curve,” Annals of Mathematics, vol. 21, no. 3, pp. 180–184, 1920. View at Publisher · View at Google Scholar · View at MathSciNet
  28. J. W. Alexander, “A proof and extension of the Jordan-Brouwer separation theorem,” Transactions of the American Mathematical Society, vol. 23, no. 4, pp. 333–349, 1922. View at Publisher · View at Google Scholar · View at MathSciNet
  29. M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, New York, NY, USA, 1961. View at MathSciNet
  30. 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 · View at MathSciNet
  31. P. A. Smith, “Book review: elements of the topology of plane sets of points,” Bulletin of the American Mathematical Society, vol. 45, no. 11, pp. 822–824, 1939. View at Publisher · View at Google Scholar · View at MathSciNet
  32. S. P. Hastings, “An existence theorem for a problem from boundary layer theory,” Archive for Rational Mechanics and Analysis, vol. 33, pp. 103–109, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. Hastings, “The existence of periodic solutions to Nagumo's equation,” The Quarterly Journal of Mathematics, vol. 25, pp. 369–378, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. S. P. Hastings, “On the existence of homoclinic and periodic orbits for the Fitzhugh-Nagumo equations,” The Quarterly Journal of Mathematics, vol. 27, no. 105, pp. 123–134, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. J. B. McLeod and J. Serrin, “The existence of similiar solutions for some laminar boundary layer problems,” Archive for Rational Mechanics and Analysis, vol. 31, pp. 288–303, 1969. View at Google Scholar · View at MathSciNet
  36. R. E. L. Turner, “Nonlinear eigenvalue problems with applications to elliptic equations,” Archive for Rational Mechanics and Analysis, vol. 42, pp. 184–193, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. P. Le Calvez, M. Martens, C. Tresser, and P. A. Worfolk, “Stably nonsynchronizable maps of the plane,” Nonlinearity, vol. 12, no. 1, pp. 9–18, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. M. Kallipoliti and P. Papasoglu, “Simply connected homogeneous continua are not separated by arcs,” Topology and Its Applications, vol. 154, no. 17, pp. 3039–3047, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. M. Henle, A Combinatorial Introduction to Topology, Dover, New York, NY, USA, 1994. View at MathSciNet
  40. K. Kuratowski, Topology. Vol. I, Academic Press, New York, NY, USA, 1966. View at MathSciNet
  41. J. G. Hocking and G. S. Young, Topology, Dover, New York, NY, USA, 2nd edition, 1988. View at MathSciNet
  42. A. Pascoletti and F. Zanolin, “A topological approach to bend-twist maps with applications,” International Journal of Differential Equations, vol. 2011, Article ID 612041, 20 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, Pa, USA, 1976. View at MathSciNet