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Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

  • Jean-Philippe Lessard
  • Jason D. Mireles James
  • Christian Reinhardt
Article

Abstract

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the Newton–Kantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.

Keywords

Computer assisted proof Invariant manifolds Parameterization method Connecting orbits Contraction mapping Transversality 

References

  1. 1.
    Ahlfors, L.V.: Complex analysis. An introduction to the theory of analytic functions of one complex variable. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill Book Co., New York (1978)Google Scholar
  2. 2.
    Arai, Z., Mischaikow, K.: Rigorous computations of homoclinic tangencies. SIAM J. Appl. Dyn. Syst. 5(2), 280–292 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    van den Berg, J.B., Lessard, J.P., Mischaikow, K., Mireles James, J.D.: Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray–Scott. SIAM J. Math. Anal. 43(4), 1557–1594 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Beyn, W.J.: The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10(3), 379–405 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J. 52(2), 283–328 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. II. Regularity with respect to parameters. Indiana Univ. Math. J. 52(2), 329–360 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. III. Overview and applications. J. Differ. Equ. 218(2), 444–515 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Capinski, M.: Covering relations and the existence of topologically normally hyperbolic invariant sets. Discret. Contin. Dyn. Syst. 23(3), 705–725 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Conley, C.: Isolated invariant sets and the Morse index. CBMS Regional Conference Series in Mathematics, American Mathematical Society, vol. 38, iii+89 (1978)Google Scholar
  10. 10.
    Coomes, B., Koçak, H., Palmer, K.: Homoclinic shadowing. J. Dyn. Differ. Equ. 17(1), 175–215 (2005)CrossRefzbMATHGoogle Scholar
  11. 11.
    Coomes, B.A., Koçak, H., Palmer, K.J.: Transversal connecting orbits from shadowing. Numer. Math. 106(3), 427–469 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Day, S., Hiraoka, Y., Mischaikow, K., Ogawa, T.: Rigorous numerics for global dynamics: a study of the Swift–Hohenberg equation. SIAM J. Appl. Dyn. Syst. 4(1), 1–31 (2005). (electronic)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Day, S., Lessard, J.-P., Mischaikow, K.: Validated continuation for equilibria of PDEs. SIAM J. Numer. Anal. 45(4), 1398–1424 (2007). (electronic)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Doedel, E.J., Friedman, M.J.: Numerical computation of heteroclinic orbits, continuation techniques and bifurcation problems. J. Comput. Appl. Math. 26(1–2), 155–170 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Doedel, E.J., Friedman, M.J., Monteiro, A.C.: On locating connecting orbits. Appl. Math. Comput. 65(1–3), 231–239 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Doedel, E.J., Friedman, M.J., Kunin, B.I.: Successive continuation for locating connecting orbits. Numer. Algorithms 14(1–3), 103–124 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Friedman, M.J., Doedel, E.J.: Numerical computation and continuation of invariant manifolds connecting fixed points. SIAM J. Numer. Anal. 28(3), 789–808 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Gidea, M., Zgliczyński, P.: Covering relations for multidimensional dynamical systems. J. Differ. Equ. 202(1), 59–80 (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    Gidea, M., Zgliczyński, P.: Covering relations for multidimensional dynamical systems. II. J. Differ. Equ. 202(1), 59–80 (2004)CrossRefzbMATHGoogle Scholar
  20. 20.
    Johnson, T., Tucker, W.: A note on the convergence of parametrised non-resonant invariant manifolds. Qual. Theory Dyn. Syst. 10(1), 107–121 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Knobloch, J., Rieß, T.: Lin’s method for heteroclinic chains involving periodic orbits. Nonlinearity 23(1), 23–54 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Koçak, H., Palmer, K., Coomes, B.: Shadowing in ordinary differential equations, Rendiconti del Seminario Matematico. Univ. Politec. Torino 65(1), 89–113 (2007)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Krauskopf, B., Rieß, T.: A Lin’s method approach to finding and continuing heteroclinic connections involving periodic orbits. Nonlinearity 21(8), 1655–1690 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Lessard, J.P., Mireles James, J.D., Reinhardt, C.: CAPSad2SadLab: numerical implementation of computer assisted proof of saddle-to-saddle connecting orbits in IntLab (2012) http://www.math.rutgers.edu/jmireles/saddleToSaddlePage.html
  25. 25.
    McCord, C., Mischaikow, K.: Connected simple systems, transition matrices, and heteroclinic bifurcations. Trans. Am. Math. Soc. 333(1), 397–422 (1992)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Mireles-James, J.D., Mischaikow, K.: Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps. Submitted (2012)Google Scholar
  27. 27.
    Mrozek, M., Żelawski, M.: Heteroclinic connections in the Kuramoto–Sivashinsky equation: a computer assisted proof. Reliab. Comput. 3(3), 277–285 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Nakao, M.T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim. 22(3–4), 321–356 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Neumaier, A., Rage, T.: Rigorous chaos verification in discrete dynamical systems. Phys. D 67(4), 327–346 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Neimark, J.I., Silnikov, L.P.: A condition for the generation of periodic motions. Doklady Akademii Nauk SSSR 160, 1261–1264 (1965)MathSciNetGoogle Scholar
  31. 31.
    Oishi, S.: Numerical verification method of existence of connecting orbits for continuous dynamical systems. J. Univ. Comput. Sci. 4(2), 193–201 (1998). (electronic)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Ortega, J.M.: The Newton–Kantorovich Theorem. Am. Math. Monthly 75, 658–660 (1968)CrossRefzbMATHGoogle Scholar
  33. 33.
    Palmer, K.J.: Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dynamics reporten. 1, 265–306, Dyn. Report. Ser. Dyn. Syst. Appl. 1, Wiley, Chichester, (1988)Google Scholar
  34. 34.
    Rudin, W.: Functional analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)Google Scholar
  35. 35.
    Rump, S.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer. 19, 287–449 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Schultz, M.H.: Spline Analysis. Prentice Hall, Upper Saddle River (1973)zbMATHGoogle Scholar
  37. 37.
    Smale, S.: Diffeomorphisms with many periodic points. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 63–80. Princeton Univ. Press, Princeton (1965)Google Scholar
  38. 38.
    Stoffer, D., Palmer, K.: Rigorous verification of chaotic behavior of maps using validated shadowing. Nonlinearity 12(6), 1683–1698 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Wilczak, D.: Abundancs of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discret. Contin. Dyn. Syst. Ser. B 11(4), 1039–1055 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Wilczak, D.: Symmetric heteroclinic connections in the Michelson system: a computer assisted proof. SIAM J. Appl. Dyn. Syst. 4(3), 489–514 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Wilczak, D.: Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discret. Contin. Dyn. Syst. Ser. B 11(4), 1039–1055 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Wilczak, D., Zgliczyński, P.: Heteroclinic connections between periodic orbits in planar restricted circular three body problem. II. Commun. Math. Phys. 259(3), 561–576 (2005)CrossRefzbMATHGoogle Scholar
  43. 43.
    Yamamoto, N.: A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed-point theorem. SIAM J. Numer. Anal. 35(5), 2004–2013 (1998). (electronic)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Zgliczyński, P.: Covering relations, cone conditions and the stable manifold theorem. J. Differ. Equ. 246(5), 1774–1819 (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jean-Philippe Lessard
    • 1
  • Jason D. Mireles James
    • 2
  • Christian Reinhardt
    • 3
  1. 1.Département de Mathématiques et de StatistiqueUniversité LavalQuébecCanada
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA
  3. 3.Technische Universität MünchenGarchingGermany

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