Now showing items 1-14 of 14

    • A survey on the inverse integrating factor 

      García, I. A. (Isaac A.); Grau Montaña, Maite (Springer, 2010)
      The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles ...
    • Averaging methods of arbitrary order, periodic solutions and integrability 

      Giné, Jaume; Llibre, Jaume; Wu, Kesheng; Zhang, Xiang (Elsevier, 2016)
      In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential ...
    • Center cyclicity for some nilpotent singularities including the Z2-equivariant class 

      García, I. A. (Isaac A.) (World Scientific Publishing, 2021-10-20)
      This work concerns with polynomial families of real planar vector fields having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal ...
    • Center cyclicity of a family of quartic polynomial differential system 

      García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Springer, 2016-09-01)
      In this paper we study the cyclicity of the centers of the quartic polynomial family written in complex notation as z = i z + z z (A z^2 + B z z + C z^2 ), where A,B,C. We give an upper bound for the cyclicity of any ...
    • Center cyclicity of Lorenz, Chen and Lü systems 

      García, I. A. (Isaac A.); Maza Sabido, Susanna; Shafer, Douglas S. (Elsevier, 2018-11-09)
      This work provides upper bounds on the cyclicity of the centers on center manifolds in the well-known Lorenz family, and also in the Chen and Lü families. We prove that at most one limit cycle can be made to bifurcate from ...
    • Cyclicity of a class of polynomial nilpotent center singularities 

      García, I. A. (Isaac A.); Shafer, Douglas S. (American Institute of Mathematical Sciences, 2016-04-01)
      In this work we extend techniques based on computational algebra for bounding the cyclicity of nondegenerate centers to nilpotent centers in a natural class of polynomial systems, those of the form $\dot x = y + P_{2m + ...
    • Cyclicity of polynomial nondegenerate centers on center manifolds 

      García, I. A. (Isaac A.); Maza Sabido, Susanna; Shafer, Douglas S. (Elsevier, 2018-11-09)
      We consider polynomial families of ordinary differential equations on $\R^3$, parametrized by the admissible coefficients, for which the origin is an isolated singularity at which the linear part of the system has one ...
    • Cyclicity of some symmetric nilpotent centers 

      García, I. A. (Isaac A.) (Elsevier, 2016)
      In this work we present techniques for bounding the cyclicity of a wide class of monodromic nilpotent singularities of symmetric polynomial planar vector fields. The starting point is identifying a broad family of nilpotent ...
    • Effective construction of Poincaré-Bendixson regions 

      Gasull, Armengol; Giacomini, Héctor; Grau Montaña, Maite (Shanghai Normal University & Wilmington Scientific Publisher, 2017)
      This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré-Bendixson regions by using transversal curves, that ...
    • Examples of center cyclicity bounds using the reduced Bautin depth 

      García, I. A. (Isaac A.) (American Mathematical Society, 2017-09-01)
      There is a method for bounding the cyclicity of nondegenerate monodromic singularities of polynomial planar families of vector fields Xλ which can work even in case that the Poincar´e first return map has associated a ...
    • The cyclicity of polynomial centers via the reduced bautin depth 

      García, I. A. (Isaac A.) (American Mathematical Society, 2016)
      We describe a method for bounding the cyclicity of the class of monodromic singularities of polyn omial planar families of vector fields X λ with an analytic Poincar e first return map having a polynomial Bautin ideal B ...
    • The Hopf cyclicity of the centers of a class of quintic polynomial vector fields 

      García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Elsevier, 2015-01-20)
      We consider families of planar polynomial vector fields having a singularity with purely imaginary eigenvalues for which a basis of its Bautin ideal B is known. We provide an algorithm for computing an upper bound of the ...
    • The inverse integrating factor and the Poincaré map 

      García, I. A. (Isaac A.); Giacomini, Héctor; Grau Montaña, Maite (American Mathematical Society, 2010)
      This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ...
    • Transversal conics and the existence of limit cycles 

      Giacomini, Héctor; Grau Montaña, Maite (Elsevier, 2015)
      This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré-Bendixson regions by using transversal conics. We present ...