American Institute of Mathematical Sciences

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

Imperfect bifurcations via topological methods in superlinear indefinite problems

 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.
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

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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
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