American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 1050-1059. doi: 10.3934/proc.2015.1050

Imperfect bifurcations via topological methods in superlinear indefinite problems

Andrea Tellini 1,

1. 

Centre d'Analyse et de Mathématiques Sociales, École des Hautes Études en Sciences Sociales, 190-198 avenue de France, 75244 Paris Cedex 13, France

Received  September 2014 Revised  March 2015 Published  November 2015

In [5] the structure of the bifurcation diagrams of a class of superlinear indefinite problems with a symmetric weight was ascertained, showing that they consist of a primary branch and secondary loops bifurcating from it. In [4] it has been proved that, when the weight is asymmetric, the bifurcation diagrams are no longer connected since parts of the primary branch and loops of the symmetric case form an arbitrarily high number of isolas. In this work we give a deeper insight on this phenomenon, studying how the secondary bifurcations break as the weight is perturbed from the symmetric situation. Our proofs rely on the approach of [5,4], i.e. on the construction of certain Poincaré maps and the study of how they vary as some of the parameters of the problems change, obtaining in this way the bifurcation diagrams.
Keywords: bifurcation diagrams, Imperfect bifurcations, superlinear indefinite problems, symmetry breaking., Poincaré maps.
Mathematics Subject Classification: 34B15, 34C23, 35B3.
Citation: Andrea Tellini. Imperfect bifurcations via topological methods in superlinear indefinite problems. Conference Publications, 2015, 2015 (special) : 1050-1059. doi: 10.3934/proc.2015.1050
References:
[1]

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

[2]

P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations,, J. Funct. Anal., 251 (2007), 573. Google Scholar

[3]

J. López-Gómez, M. Molina-Meyer and A. Tellini, Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics,, DCDS Supplement 2013 Proceedings of the 9th AIMS International Conference, (2013), 515. Google Scholar

[4]

J. López-Gómez and A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem,, Nonlinear Anal. - Theory Methods Appl., 108 (2014), 223. Google Scholar

[5]

J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems,, Commun. Pure Appl. Anal., 13 (2014), 1. Google Scholar

show all references

References:
[1]

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

[2]

P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations,, J. Funct. Anal., 251 (2007), 573. Google Scholar

[3]

J. López-Gómez, M. Molina-Meyer and A. Tellini, Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics,, DCDS Supplement 2013 Proceedings of the 9th AIMS International Conference, (2013), 515. Google Scholar

[4]

J. López-Gómez and A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem,, Nonlinear Anal. - Theory Methods Appl., 108 (2014), 223. Google Scholar

[5]

J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems,, Commun. Pure Appl. Anal., 13 (2014), 1. Google Scholar

[1]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Spiraling bifurcation diagrams in superlinear indefinite problems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1561-1588. doi: 10.3934/dcds.2015.35.1561
[2]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515
[3]

Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1
[4]

Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112
[5]

Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103
[6]

Anna Goƚȩbiewska, Norimichi Hirano, Sƚawomir Rybicki. Global symmetry-breaking bifurcations of critical orbits of invariant functionals. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2005-2017. doi: 10.3934/dcdss.2019129
[7]

Lucio Cadeddu, Giovanni Porru. Symmetry breaking in problems involving semilinear equations. Conference Publications, 2011, 2011 (Special) : 219-228. doi: 10.3934/proc.2011.2011.219
[8]

Claudia Anedda, Giovanni Porru. Symmetry breaking and other features for Eigenvalue problems. Conference Publications, 2011, 2011 (Special) : 61-70. doi: 10.3934/proc.2011.2011.61
[9]

M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure & Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411
[10]

Julián López-Gómez, Pavol Quittner. Complete and energy blow-up in indefinite superlinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 169-186. doi: 10.3934/dcds.2006.14.169
[11]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505
[12]

Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047
[13]

Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014
[14]

C. Bandle, Y. Kabeya, Hirokazu Ninomiya. Imperfect bifurcations in nonlinear elliptic equations on spherical caps. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1189-1208. doi: 10.3934/cpaa.2010.9.1189
[15]

Fanni M. Sélley. Symmetry breaking in a globally coupled map of four sites. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3707-3734. doi: 10.3934/dcds.2018161
[16]

Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886
[17]

E. Kapsza, Gy. Károlyi, S. Kovács, G. Domokos. Regular and random patterns in complex bifurcation diagrams. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 519-540. doi: 10.3934/dcdsb.2003.3.519
[18]

Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297
[19]

Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436
[20]

Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333

 Impact Factor: 

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences