Now showing items 1-20 of 40

    • A method for characterizing nilpotent centers 

      Giné, Jaume; Llibre, Jaume (Elsevier, 2014)
      To characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest, first for the problem of distinguishing between a focus and a center, and after for studying the ...
    • A note on Liouvillian first integrals and invariant algebraic curves 

      Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (Elsevier, 2013-02)
      In this work we study the existence and non-existence of finite invariant algebraic curves for a complex planar polynomial differential system having a Liouvillian first integral.
    • A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter 

      Buica, Adriana; Giné, Jaume; Llibre, Jaume (Elsevier, 2012)
      We deal with nonlinear T–periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to ...
    • A sufficient condition in order that the real Jacobian conjecture in R^2 holds 

      Braun, Francisco; Giné, Jaume; Llibre, Jaume (Elsevier, 2016)
      Let F=(f,g):R2→R2 be a polynomial map such that det⁡DF(x,y) is different from zero for all (x,y)∈R2 and F(0,0)=(0,0). We prove that for the injectivity of F it is sufficient to assume that the higher homogeneous terms of ...
    • Analytic nilpotent centers as limits of nondegenerate centers revisited 

      García, I. A. (Isaac A.); Giacomini, Héctor; Giné, Jaume; Llibre, Jaume (Elsevier, 2016-04-25)
      We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré--Liapunov method to detect centers with purely imaginary ...
    • Analytic reducibility of nondegenerate centers: Cherkas systems 

      Giné, Jaume; Llibre, Jaume (Bolyai Institute, University of Szeged; Hungarian Academy of Sciences, 2016)
      In this paper we study the center problem for polynomial differential systems and we prove that any center of an analytic differential system is analytically reducible. x˙=y,y˙=P0(x)+P1(x)y+P2(x)y2, We also study the ...
    • 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 ...
    • Averaging theory at any order for computing periodic orbits 

      Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (Elsevier, 2013-05)
      We provide a recurrence formula for the coefficients of the powers of ε in the series expansion of the solutions around ε=0 of the perturbed first-order differential equations. Using it, we give an averaging theory at any ...
    • 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 ...
    • Centers and isochronous centers for generalized quintic systems 

      Giné, Jaume; Llibre, Jaume; Valls, Claudia (Elsevier, 2015)
      In this paper we classify the centers and the isochronous centers of certain polynomial differential systems in R2 of degree d≥5 odd that in complex notation are ż=(λ+i)z(zz̄)d−52(Az5+Bz4z̄+Cz3z̄2+Dz2z̄3+Ezz̄4+Fz̄5), ...
    • Centers for a class of generalized quintic polynomial differential systems 

      Giné, Jaume; Llibre, Jaume; Valls, Claudia (Elsevier, 2014)
      We classify the centers of the polynomial differential systems in R2 of degree d ≥ 5 odd that in complex notation writes as z˙ = iz + (zz¯)d−5/2 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5), where A, B, C, D, E, F ∈ C ...
    • Centers for generalized quintic polynomial differential systems 

      Giné, Jaume; Llibre, Jaume; Valls, Claudia (Rocky Mountain Mathematics Consortium, 2017)
      We classify the centers of polynomial differential systems in $R^2$ of odd degree $d \ge 5$, in complex notation, as $\dot{z} = iz + (z \bar z)^(d-5)/2(A z^5 + B z^4 \bar z + C z^3 \bar z^2 + D z^2 \bar z^3 + E z \bar z^4 ...
    • Centers for the Kukles homogeneous systems with even degree 

      Giné, Jaume; Llibre, Jaume; Valls, Claudia (Shanghai Normal University & Wilmington Scientific Publisher, 2017)
      For the polynomial differential system x˙=−y, y˙=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n≥2 has a ...
    • Centers for the Kukles homogeneous systems with odd degree 

      Giné, Jaume; Llibre, Jaume; Valls, Claudia (London Mathematical Society, 2015)
      For the polynomial differential system x˙ = −y, y˙ = x+Qn(x; y), where Qn(x; y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for ...
    • Centers of weight-homogeneous polynomial vector fields on the plane 

      Giné, Jaume; Llibre, Jaume; Valls, Claudia (American Mathematical Society, 2017)
      We characterize all centers of a planar weight-homogeneous polynomial vector fields. Moreover we classify all centers of a planar weight-homogeneous polynomial vector fields of degrees $6$ and $7$.
    • Chiellini Hamiltonian Liénard differential systems 

      Giné, Jaume; Llibre, Jaume; Valls, Claudia (Texas State University, 2019)
      We characterize the centers of the Chiellini Hamiltonian Li´enard second-order differential equations x 0 = y, y 0 = −f(x)y − g(x) where g(x) = f(x)(k − α(1 + α) R f(x)dx) with α, k ∈ R. Moreover we study the ...
    • Classical planar algebraic curves realizable by quadratic polynomial differential systems 

      García, I. A. (Isaac A.); Llibre, Jaume (World Scientific Publishing, 2017-11-15)
      In this paper we show planar quadratic polynomial differentialsystems exhibiting as solutions some famous planar invariant algebraic curves. Also we put particular attention to the Darboux integrability of these differential ...
    • Cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors 

      García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Elsevier, 2013)
      First we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus. After we use this vanishing multiplicity for studying the cyclicity ...
    • Divergence and Poincare-Liapunov constants for analytic differential systems 

      Grau Montaña, Maite; Llibre, Jaume (Elsevier, 2015)
      We consider a planar autonomous real analytic differential system with a monodromic singular point p. We deal with the center problem for the singular point p. Our aim is to highlight some relations between the divergence ...
    • Integrability conditions of a resonant saddle in generalized Liénard-like complex polynomial differential systems 

      Giné, Jaume; Llibre, Jaume (Elsevier, 2017)
      We consider a complex differential system with a resonant saddle at the origin. We compute the resonant saddle quantities and using Gröbner bases we find the integrability conditions for such systems up to a certain degree. ...