Global secondary bifurcation, symmetry breaking and period-doubling

Rainer Mandel

DOI: http://dx.doi.org/10.12775/TMNA.2019.025

Abstract


In this paper we provide a criterion for global secondary bifurcation via symmetry breaking. As an application, the occurrence of period-doubling bifurcations for the Lugiato-Lefever equation is proved.

Keywords


Secondary bifurcation; global bifurcation; symmetry breaking; period-doubling bifurcation; Lugiato--Lefever equation

Full Text:

PREVIEW FULL TEXT

References


A. Ambrosetti and A. Malchiodi, Nonlinear analysis and semilinear elliptic problems, Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge, 2007.

T. Bartsch, E.N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations 37 (2010), 345–361.

T. Bartsch, R. Tian and Z.-Q. Wang, Bifurcations for a coupled Schrödinger system with multiple components, Z. Angew. Math. Phys. 66 (2015), no. 5, 2109–2123.

L. Bauer, H.B. Keller and E.L. Reiss, Multiple eigenvalues lead to secondary bifurcation 17 (1975), 101–122.

R. Böhme, Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme, Math. Z. 127 (1972), 105–126.

J. Bracho, M. Clapp and W. Marzantowicz, Symmetry breaking solutions of nonlinear elliptic systems, Topol. Methods Nonlinear Anal. 26 (2005), 189–201.

G. Cerami, Symmetry breaking for a class of semilinear elliptic problems, Nonlinear Anal. 10 (1986), 1–14.

C.V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differential Equations 54 (1984), no. 3, 429–437.

M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340.

E.N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J. 23 (1973/74), 1069–1076.

E.N. Dancer, Breaking of symmetries for forced equations, Math. Ann. 262 (1983), no. 4, 473–486.

E.N. Dancer, Global breaking of symmetry of positive solutions on two-dimensional annuli, Differential Integral Equations 5 (1992), no. 4, 903–913.

K. Deimling, Nonlinear functional analysis, Springer–Verlag, Berlin, 1985.

B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243.

H. Kielhöfer, Bifurcation Theory, Applied Mathematical Sciences, vol. 156, second ed., Springer, New York, 2012.

M.A. Krasnosel’skiı̆, Topological Methods in the Theory of NonlinearIintegral Equations, The Macmillan Co., New York, 1964.

K. Kuto, T. Mori, T. Tsujikawa and S. Yotsutani, Secondary bifurcation for a nonlocal Allen–Cahn equation, J. Differential Equations 263 (2017), no. 5, 2687–2714.

S.-S. Lin, On non-radially symmetric bifurcation in the annulus, J. Differential Equations 80 (1989), no. 2, 251–279.

L.A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett. 58, no. 21, 2209–2211.

R. Mandel, Grundzustände, Verzweigungen und singuläre Lösungen nichtlinearer Schrödingersysteme, Karlsruhe Institute of Technology (KIT), 2013.

R. Mandel and W. Reichel, A priori bounds and global bifurcation results for frequency combs modeled by the Lugiato–Lefever equation, SIAM J. Appl. Math. 77 (2017), 315–345.

A. Marino, La biforcazione nel caso variazionale, Confer. Sem. Mat. Univ. Bari 132 (1973), pp. 14.

J. Mawhin, Leray–Schauder degree: a half century of extensions and applications, Topol. Methods Nonlinear Anal. 14 (1999), no. 2, 195–228.

M. Miyamoto, Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry, J. Math. Anal. Appl. 357 (2009), 89–97.

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.

J. Smoller and A.G. Wasserman, Bifurcation and symmetry-breaking, Invent. Math. 100 (1990), no. 1, 63–95.

P.N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 2, 107–112.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism