Advertisement

Horseshoes in 3D equations with applications to Lotka–Volterra systems

  • Alfonso Ruiz-Herrera
  • Fabio Zanolin
Article

Abstract

We discuss a geometric configuration for a class of homeomorphisms in \({\mathbb{R}^3}\) producing the existence of infinitely many periodic points as well a complex dynamics due to the presence of a topological horseshoe. We also show that such a class of homeomorphisms appears in the classical Lotka–Volterra system.

Mathematics Subject Classification

Primary 34C25 Secondary 37C25 37D45 

Keywords

Complex dynamics Poincaré’s map Topological horseshoes Periodic solutions Three dimensional predator–prey equations Switching systems 

References

  1. 1.
    Ahmad S., Lazer A.C.: Average growth and extinction in a competitive Lotka–Volterra system. Nonlinear Anal. 62, 545–557 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arioli G., Zgliczyński P.: Symbolic dynamics for the Hénon–Heiles Hamiltonian on the critical level. J. Differ. Equ. 171, 173–202 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Block, L.S., Coppel, W.A.: Dynamics in One Dimension, Lecture Notes in Mathematics, vol. 1513. Springer, Berlin, New York (1992)Google Scholar
  4. 4.
    Burns K., Weiss H.: A geometric criterion for positive topological entropy. Commun. Math. Phys. 172, 95–118 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Capietto A., Dambrosio W., Papini D.: Superlinear indefinite equations on the real line and chaotic dynamics. J. Differ. Equ. 181, 419–438 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carbinatto M., Kwapisz J., Mischaikow K.: Horseshoes and the Conley index spectrum. Ergod. Theory Dyn. Syst. 20, 365–377 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cushing J.M.: Periodic time-dependent predator–prey systems. SIAM J. Appl. Math. 32, 82–95 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cushing J.M.: Periodic two-predator, one-prey interactions and the time sharing of a resource niche. SIAM J. Appl. Math. 44, 392–410 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    DuBowy P.J.: Waterfowl communities and seasonal environments: temporal variability in interspecific competition. Ecology 69, 1439–1453 (1988)CrossRefGoogle Scholar
  10. 10.
    Gaines, R.E., Mawhin, J.L.: Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568. Springer, Berlin, New York (1977)Google Scholar
  11. 11.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Revised and Corrected Reprint of the 1983 Original. Applied Mathematical Sciences, 42. Springer, New York (1990)Google Scholar
  12. 12.
    Hsu S.-B., Zhao X.-Q.: A Lotka–Volterra competition model with seasonal succession. J. Math. Biol. 64, 109–130 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hu S.S., Tessier A.J.: Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage. Ecology 76, 2278–2294 (1995)CrossRefGoogle Scholar
  14. 14.
    Huppert A., Blasius B., Olinky R., Stone L.: A model for seasonal phytoplankton blooms. J. Theor. Biol. 236, 276–290 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Keeling, M., Rohani, P., Grenfell, P., B.T.: Seasonally forced disease dynamics explored as switching between attractors. Physica D 148, 317–335 (2001)Google Scholar
  16. 16.
    Kennedy J., Koçak S., Yorke J.A.: A chaos lemma. Am. Math. Mon. 108, 411–423 (2001)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kennedy J., Yorke J.A.: Topological horseshoes. Trans. Am. Math. Soc. 353, 2513–2530 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kennedy J., Yorke J.A.: Generalized Hénon difference equations with delay. Univ. Iagel. Acta Math. 41, 9–28 (2003)MathSciNetGoogle Scholar
  19. 19.
    Kirchgraber U., Stoffer D.: On the definition of chaos. Z. Angew. Math. Mech. 69, 175–185 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Krasnosel’skiĭ, M.A.: The Operator of Translation Along the Trajectories of Differential Equations, Translations of Mathematical Monographs, vol. 19. American Mathematical Society, Providence (1968)Google Scholar
  21. 21.
    Krasnosel’skiĭ M.A., Zabreĭko, P.P.: Geometrical methods of nonlinear analysis, Grundlehren der Mathematischen Wissenschaften 263, Springer, Berlin (1984)Google Scholar
  22. 22.
    Krikorian N.: The Volterra model for three species predator–prey systems: boundedness and stability. J. Math. Biol. 7, 117–132 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lisena B.: Asymptotic behaviour in periodic three species predator–prey systems. Ann. Mat. Pura Appl. 186, 85–98 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Malik T., Smith H.L.: Does dormancy increase fitness of bacterial populations in time-varying environments?. Bull. Math. Biol. 70, 1140–1162 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mawhin, J.: Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, 40. American Mathematical Society, Providence (1979)Google Scholar
  26. 26.
    Mawhin, J.: The legacy of Pierre–François Verhulst and Vito Volterra in population dynamics. In: The first 60 years of nonlinear analysis of Jean Mawhin, pp. 147–160. World Science Publications, River Edge (2004)Google Scholar
  27. 27.
    Medio A., Pireddu M., Zanolin F.: Chaotic dynamics for maps in one and two dimensions: a geometrical method and applications to economics. Int. J. Bifur. Chaos Appl. Sci. Eng. 19, 3283–3309 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Moser, J.: Stable and random motions in dynamical systems. With Special Emphasis on Celestial Mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. Annals of Mathematics Studies, No. 77. Princeton University Press, Princeton (1973)Google Scholar
  29. 29.
    Papini D., Zanolin F.: On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill’s equations. Adv. Nonlinear Stud. 4, 71–91 (2004)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Papini D., Zanolin F.: Some results on the periodic points and chaotic dynamics arising from the study of the nonlinear Hill equation. Rend. Sem. Mat. Univ. Pol. Torino (Subalpine Rhapsody in Dynamics) 65, 115–157 (2007)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Pireddu, M., Zanolin, F.: Cutting surfaces and applications to periodic points and chaotic-like dynamics. Topol. Methods Nonlinear Anal. 30, 279–319 (2007); Correction in Topol. Methods Nonlinear Anal. 33, 395 (2009)Google Scholar
  32. 32.
    Rouche, N., Mawhin, J.: Équations différentielles ordinaires. Tome II: Stabilité et solutions périodiques. Masson et Cie, Paris (1973)Google Scholar
  33. 33.
    Ruiz-Herrera, A.: Chaos in predatorprey systems with/without impulsive effect. Nonlinear Anal. Real World Appl. 13, 977–986 (2012)Google Scholar
  34. 34.
    Ruiz-Herrera A., Zanolin F.: An example of chaotic dynamics in 3D systems via stretching along paths. Ann. Mat. Pura Appl. 193, 163–185 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Smale, S. Diffeomorphisms with many periodic points. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) pp. 63–80. Princeton University Press, Princeton (1965)Google Scholar
  36. 36.
    Smale S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Smale S.: Finding a horseshoe on the beaches of Rio. Math. Intell. 20, 39–44 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Srzednicki, R., Wójcik, K., Zgliczyński, P.: Fixed point results based on the Ważewski method, In: Brown, R.F., Furi, M., Górniewicz, L., Jiang, B. (eds.) Handbook of Topological Fixed Point Theory, pp. 905–943. Springer, Dordrecht (2005)Google Scholar
  39. 39.
    Volterra V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie R. Accad. Lincei 6(2), 31–113 (1926)Google Scholar
  40. 40.
    Waldvogel J.: The period in the Lotka–Volterra system is monotonic. J. Math. Anal. Appl. 114, 178–184 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wójcik K.: Remark on complicated dynamics of some planar system. J. Math. Anal. Appl. 271, 257–266 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zgliczyński P.: Fixed point index for iterations of maps. topological horseshoe and chaos. Topol. Methods Nonlinear Anal. 8, 169–177 (1986)Google Scholar
  43. 43.
    Zgliczyński P., Gidea M.: Covering relations for multidimensional dynamical systems. J. Differ. Equ. 202, 32–58 (2004)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada IIUniversidade de VigoVigoSpain
  2. 2.Department of Mathematics and Computer ScienceUniversity of UdineUdineItaly

Personalised recommendations