Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios

  • Jürgen Knobloch
  • David J. B. Lloyd
  • Björn Sandstede
  • Thomas Wagenknecht
Article

Abstract

Homoclinic snaking refers to the bifurcation structure of symmetric localised roll patterns that are often found to lie on two sinusoidal “snaking” bifurcation curves, which are connected by an infinite number of “rung” segments along which asymmetric localised rolls of various widths exist. The envelopes of all these structures have a unique maximum and we refer to them as symmetric or asymmetric 1-pulses. In this paper, the existence of stationary 1D patterns of symmetric 2-pulses that consist of two well-separated 1-pulses is established. Corroborating earlier numerical evidence, it is shown that symmetric 2-pulses exist along isolas in parameter space that are formed by parts of the snaking curves and the rungs mentioned above.

Keywords

Homoclinic snaking Isolas Multi-pulses Swift–Hohenberg equation 

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References

  1. 1.
    Avitabile D., Lloyd D.J.B., Burke J., Knobloch E., Sandstede B.: To snake or not to snake in the planar Swift–Hohenberg equation. SIAM J. Appl. Dynam. Syst. 9, 704–733 (2010)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beck M., Knobloch J., Lloyd D.J.B., Sandstede B., Wagenknecht T.: Snakes, ladders, and isolas of localised patterns. SIAM J. Math. Anal. 41, 936–972 (2009)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Belitskii G.R., Kopanskii A.Y.: Sternberg theorem for equivariant Hamiltonian vector fields. Nonlinear Anal. 47, 4491–4499 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burke J., Knobloch E.: Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73, 056211 (2006)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Burke J., Houghton S.M., Knobloch E.: Swift–Hohenberg equation with broken reflection symmetry. Phys. Rev. E 80, 036202 (2009)CrossRefGoogle Scholar
  6. 6.
    Burke, J., Knobloch, E.: Multipulse states in the Swift–Hohenberg equation. In: Hou X., Lu X., Miranville A., Su J., Zhu J. (eds.) Dynamical Systems and Differential Equations, pp. 109-117 (2009)Google Scholar
  7. 7.
    Champneys A.R., Kirk V., Knobloch E., Oldeman B.E., Rademacher J.D.M.: Unfolding a tangent equilibrium-to-periodic heteroclinic cycle. SIAM J. Appl. Dyn. Syst. 8, 1261–1304 (2009)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Champneys A.R., Toland J.F.: Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems. Nonlinearity 6, 665–721 (1993)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chapman S.J., Kozyreff G.: Exponential asymptotics of localised patterns and snaking bifurcation diagrams. Physica D 238, 319–354 (2009)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Coullet P., Riera C., Tresser C.: Stable static localized structures in one dimension. Phys. Rev. Lett. 84, 3069–3072 (2000)CrossRefGoogle Scholar
  11. 11.
    Dawes J.H.P.: The emergence of a coherent structure for coherent structures: localized states in nonlinear systems. Phil. Trans. R. Soc. A 368, 3519–3534 (2010)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Devaney R.L.: Homoclinic orbits in Hamiltonian systems. J. Differ. Equ. 21, 431–438 (1976)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Härterich J.: Cascades of reversible homoclinic orbits to a saddle-focus equilibrium. Physica D 112, 187–200 (1998)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    van der Heijden G.H.M., Champneys A.R., Thompson J.M.T.: Spatially complex localisation in twisted elastic rods constrained to a cylinder. Int. J. Solids Struct. 39, 1863–1883 (2002)MATHCrossRefGoogle Scholar
  15. 15.
    Houghton, S.M., Wagenknecht, T.: Multi-pulses in the Swift–Hohenberg equation with broken symmetry (2010) (in preparation)Google Scholar
  16. 16.
    Knobloch E.: Spatially localized structures in dissipative systems: open problems. Nonlinearity 21, T45–T60 (2008)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Knobloch, J., Rieß, T., Vielitz, M.: Nonreversible homoclinic snaking (2010) (preprint)Google Scholar
  18. 18.
    Kozyreff G., Chapman S.J.: Asymptotics of large bound states of localised structures. Phys. Rev. Lett. 97, 044502 (2006)CrossRefGoogle Scholar
  19. 19.
    Lerman L.M.: Dynamical phenomena near a saddle-focus homoclinic connection in a Hamiltonian system. J. Stat. Phys. 101, 357–372 (2000)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Moser J.: On the generalisation of a theorem of A. Liapounoff. Comm. Pure Appl. Math. 11, 257–271 (1958)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Oldeman B.E., Champneys A.R., Krauskopf B.: Homoclinic branch switching: a numerical implementation of Lin’s method. Int. J. Bif. Chaos 13, 2977–3000 (2003)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Sandstede B.: Stability of multiple-pulse solutions. Trans. Am. Math. Soc. 350, 429–472 (1998)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sandstede B.: Instability of localized buckling modes in a one-dimensional strut model. Philos. Trans. R. Soc. Lond. A 355, 2083–2097 (1997)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wadee M.K., Coman C.D., Bassom A.P.: Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation. Physica D 163, 26–48 (2002)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Woods P.D., Champneys A.R.: Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation. Physica D 129, 147–170 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Jürgen Knobloch
    • 1
  • David J. B. Lloyd
    • 2
  • Björn Sandstede
    • 3
  • Thomas Wagenknecht
    • 4
  1. 1.Department of MathematicsTechnical University IlmenauIlmenauGermany
  2. 2.Department of MathematicsUniversity of SurreyGuildfordUK
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA
  4. 4.Department of Applied MathematicsUniversity of LeedsLeedsUK

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