A new result on averaging theory for a class of discontinuous planar differential systems with applications

• Jackson Itikawa ^[1] ; Jaume Llibre ^[2] ; Douglas Duarte Novaes ^[3]
  1. [1] Universidade de São Paulo

     Universidade de São Paulo

     Brasil

     
  2. [2] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  3. [3] Universidade Estadual de Campinas

     Universidade Estadual de Campinas

     Brasil

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 33, Nº 4, 2017, págs. 1247-1265
• Idioma: inglés
• DOI: 10.4171/RMI/970
• Texto completo no disponible (Saber más ...)
• Resumen

  • We develop the averaging theory at any order for computing the periodic solutions of periodic discontinuous piecewise differential system of the form drdθ=r′={F+(θ,r,ϵ)if0≤θ≤α,F−(θ,r,ϵ)ifα≤θ≤2π, where F±(θ,r,ϵ)=∑ki=1ϵiF±i(θ,r)+ϵk+1R±(θ,r,ϵ) with θ∈S1 and r∈D, where D is an open interval of R+, and ϵ is a small real parameter.

    Applying this theory, we provide lower bounds for the maximum number of limit cycles that bifurcate from the origin of quartic polynomial differential systems of the form x˙=−y+xp(x,y),y˙=x+yp(x,y), with p(x,y) a polynomial of degree 3 without constant term, when they are perturbed, either inside the class of all continuous quartic polynomial differential systems, or inside the class of all discontinuous piecewise quartic polynomial differential systems with two zones separated by the straight line y=0.