BIT Numerical Mathematics

, Volume 56, Issue 2, pp 605–631 | Cite as

Nonautonomous systems with transversal homoclinic structures under discretization

  • Alina Girod
  • Thorsten Hüls


We consider homoclinic orbits in continuous time nonautonomous dynamical systems. Unlike the autonomous case, stable and unstable fiber bundles that generalize stable and unstable manifolds, typically intersect transversally in isolated points. In the first part, we establish persistence and error estimates for one-step discretizations of transversal homoclinic orbits. Secondly, we extend an algorithm by England, Krauskopf, Osinga to nonautonomous systems and illustrate transversally intersecting fibers along homoclinic orbits for three examples. The first one is constructed artificially in order to study numerical errors, while the second one is a periodically forced model that reveals the influence of underlying autonomous dynamics. The third example originates from mathematical biology.


Nonautonomous dynamical systems Homoclinic orbits  Fiber bundles Discretization effects Approximation theory Exponential dichotomy 

Mathematics Subject Classification

70K44 55R10 37B55 34C37 



The authors thank an anonymous referee for constructive and helpful remarks, which improved the first version of this paper.


  1. 1.
    Aulbach, B.: The fundamental existence theorem on invariant fiber bundles. J. Differ. Equ. Appl. 3(5–6), 501–537 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Beyn, W.-J.: The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10(3), 379–405 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beyn, W.-J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov, Y., Sandstede, B.: Chapter 4 numerical continuation, and computation of normal forms. Handbook of Dynamical Systems, vol. 2, pp. 149–219. Elsevier, Amsterdam (2002)Google Scholar
  4. 4.
    Beyn, W.-J., Kleinkauf, J.-M.: The numerical computation of homoclinic orbits for maps. SIAM J. Numer. Anal. 34, 1207–1236 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Coppel, W.: Dichotomies in Stability Theory. Lecture notes in mathematics, vol. 629. Springer-Verlag, New York (1978)zbMATHGoogle Scholar
  6. 6.
    England, J., Krauskopf, B., Osinga, H.: Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse. SIAM J. Appl. Dyn. Syst. 3(2), 161–190 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fiedler, B., Scheurle, J.: Discretization of homoclinic orbits, rapid forcing and “invisible” chaos. Mem. Am. Math. Soc. 119(570), viii+79 (1996)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Friedman, M., Doedel, E.: Computational methods for global analysis of homoclinic and heteroclinic orbits: a case study. J. Dyn. Differ. Equ. 5(1), 37–57 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garay, B.: On \(C^j\)-closeness between the solution flow and its numerical approximation. J. Differ. Equ. Appl. 2(1), 67–86 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  11. 11.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture notes in mathematics, vol. 840. Springer, Berlin (1981)zbMATHGoogle Scholar
  12. 12.
    Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Number Nr. 583 in lecture notes in mathematics. Springer, Berlin (1997)zbMATHGoogle Scholar
  13. 13.
    Hüls, T.: Homoclinic orbits of non-autonomous maps and their approximation. J. Differ. Equ. Appl. 12(11), 1103–1126 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hüls, T.: Numerical computation of dichotomy rates and projectors in discrete time. Discrete Contin. Dyn. Syst. Ser. B 12(1), 109–131 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hüls, T.: Computing Sacker–Sell spectra in discrete time dynamical systems. SIAM J. Numer. Anal. 48(6), 2043–2064 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hüls, T.: Homoclinic trajectories of non-autonomous maps. J Differ. Equ. Appl 17(1), 9–31 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hüls, T.: A contour algorithm for computing stable fiber bundles of nonautonomous, noninvertible maps. Bielefeld University, CRC 701. Preprint 14060, (2014)Google Scholar
  18. 18.
    Irwin, M.: Smooth dynamical systems. Advanced Series in Nonlinear Dynamics, vol. 17. World Scientific Publishing Co., Inc, River Edge (2001). Reprint of the 1980 original, With a foreword by R. S, MacKayGoogle Scholar
  19. 19.
    Kleinkauf, J.-M.: The numerical computation and geometrical analysis of heteroclinic tangencies. Technical Report 98-052, Bielefeld University, SFB 343. (1998)
  20. 20.
    Krauskopf, B., Osinga, H.: Growing \(1\)D and quasi-\(2\)D unstable manifolds of maps. J. Comput. Phys. 146(1), 404–419 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Langa, J., Robinson, J., Suárez, A.: Stability, instability, and bifurcation phenomena in non-autonomous differential equations. Nonlinearity 15(3), 887–903 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lerman, L., Shiínikov, L.: Homoclinical structures in nonautonomous systems: nonautonomous chaos. Chaos 2(3), 447–454 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Palmer, K.: Exponential dichotomies and transversal homoclinic points. J. Differ. Equ. 55(2), 225–256 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Palmer, K.: Exponential dichotomies, the shadowing lemma and transversal homoclinic points. In: Dynamics reported, vol. 1 of dynamics report. Serious dynamic systems application, pp. 265–306. Wiley, Chichester, (1988)Google Scholar
  25. 25.
    Pötzsche, C., Rasmussen, M.: Taylor approximation of integral manifolds. J. Dyn. Differ. Equ. 18(2), 427–460 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Scheffer, M., Rinaldi, S., Kuznetsov, Y., van Nes, E.: Seasonal dynamics of daphnia and algae explained as a periodically forced predator-prey system. Oikos 80, 519–532 (1997)CrossRefGoogle Scholar
  27. 27.
    Šil’nikov, L.: Existence of a countable set of periodic motions in a neighborhood of a homoclinic curve. Dokl. Akad. Nauk SSSR, 172(1967), 298–301. Soviet Math. Dokl. 8, 102–106 (1967)Google Scholar
  28. 28.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Vainikko, G.: Approximative methods for nonlinear equations (two approaches to the convergence problem). Nonlinear Anal. Theory Methods Appl. 2(6), 647–687 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zou, Y.-K., Beyn, W.-J.: On manifolds of connecting orbits in discretizations of dynamical systems. Nonlinear Anal. Theory Methods Appl. 52(5), 1499–1520 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsBielefeld UniversityBielefeldGermany

Personalised recommendations