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Abstract
This article deals with the bifurcation of polycycles and limit cycles within the 1parameter families of planar vector fields X_m^k, defined by =y^3x^2k 1,=x my^4k 1, where m is a real parameter and k1 integer. The bifurcation diagram for the separatrix skeleton of X_m^k in function of m is determined and the one for the global phase portraits of (X^1_m)_mR is completed. Furthermore for arbitrary k1 some bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of X_m^k is found to be uniformly bounded independent of mR and the Hilbert number for (X_m^k)_mR, that thus is finite, is found to be at least one.
Original language  English 

Pages (fromto)  09891013 
Number of pages  25 
Journal  Journal of Differential Equations 
Volume  259 
DOIs  
Publication status  Accepted in press  2015 
Keywords
 Global phase portrait
 Hilbert's 16th Problem
 Limit cycles
 Nilpotent center problem
 Rotated vector field
 Separatrix skeleton
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Sistemas dinámicos contínuos y discretos: bifurcaciones, orbitas periódicas, integrabilidad y aplicaciones
Torregrosa i Arús, J. (Principal Investigator), Garrido Pelaez, J. M. (Collaborator), Jimenez Ruiz, J. J. (Collaborator), Mañosas Capellades, F. (Collaborator), Artes Ferragud, J. C. (Investigator), Cima Mollet, A. M. (Investigator), Corbera Subirana, M. (Investigator), Cors Iglesias, J. M. (Investigator), Gasull Embid, A. (Investigator), Mañosa Fernández, V. (Investigator), Pantazi, C. (Investigator) & Mayayo Cortasa, T. (Collaborator)
1/09/23 → 31/08/26
Project: Research Projects and Other Grants