Phase portraits of linear type centers of polynomial hamiltonian systems with hamiltonian function of degree 5 of the form H = H<inf>1(x)</inf> + H<inf>2(y)</inf>

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal

Research output: Contribution to journalArticleResearch

1 Citation (Scopus)

Abstract

© Copyright 2018 American Institute of Mathematical Sciences. We study the phase portraits on the Poincaré disc for all the linear type centers of polynomial Hamiltonian systems of degree 5 with Hamiltonian function H(x, y) = H1(x) + H2(y), where H1(x) = 1/2 x2 + a3/3 x3 + a4/4 x4 + a5/5 x5 and H2(y) = 1/2 y2 + b3/3 y3 + b4/4 y4 + b5/5 y5 as function of the six real parameters a3, a4, a5, b3, b4 and b5 with a5b5 ≠ 0. We characterize the type and multiplicity of the roots of the polynomials p(y) = 1 + b3y + b4y2 + b5y3 and q(x) = 1 + a3x + a4x2 + a5x3 and we prove that the finite equilibria are saddles, centers, cusps or the union of two hyperbolic sectors. For the infinite equilibria we found that there only exist two nodes on the Poincaré disc with opposite stability. We also characterize the separatrices of the equilibria and analyze the possible connections between them. As a complement we use the energy level to complete the global phase portrait.
Original languageEnglish
Pages (from-to)75-113
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume39
DOIs
Publication statusPublished - 1 Jan 2019

Keywords

  • Linear type centers
  • Phase portraits
  • Quartic vector field.
  • Separable Hamiltonian systems

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