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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 883–909 | Cite as

Analytic Tools to Bound the Criticality at the Outer Boundary of the Period Annulus

  • F. Mañosas
  • D. Rojas
  • J. Villadelprat
Article

Abstract

In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most \(n\geqslant 0\) critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family \(\ddot{x}=x^p-x^q,\) \(p,q\in {\mathbb {R}}\) with \(p>q\).

Keywords

Center Period function Critical periodic orbit Bifurcation Criticality Chebyshev system 

Mathematics Subject Classification

34C07 34C23 34C25 

Notes

Acknowledgments

All the authors are partially supported by the MINECO Grant MTM2014-52209-C2-1-P. D. Rojas is also partially supported by FI-DGR 2014 of Generalitat de Catalunya

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

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