KAM Tori for a Two Dimensional Beam Equation with aQuintic Nonlinear Term and Quasi-periodic Forcing

• Min Zhang ^[1] ; Jie Rui ^[1] ; Yan Li ^[1] ; Jian Zhang ^[1]
  1. [1] China University of Petroleum (East China)
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
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• Resumen

  • This work studies a two-dimensional beam equation with a quintic nonlinear term andquasi-periodic forcingutt+2u+εφ(t)h(u)=0,x∈T2,t∈Rwith periodic boundary conditions, whereεis a small positive parameter;φ(t)is a realanalytic quasi-periodic function intwith frequency vectorη=(η1,η2...,ηn∗)⊂[,2]n∗for a given positive integern∗and some constant>0; andhis a realanalytic function of the formh(u)=c1u+c5u5+∑ˆi≥6cˆiuˆi,c1,c5=0.Firstly, the linear part of Hamiltonian system corresponding to the equation is trans-formed to constant coefficients by a linear quasi-periodic change of variables. Then,a symplectic transformation is used to convert the Hamiltonian system into an angle-dependent block-diagonal normal form, which can be achieved by selecting theappropriate tangential sites. Finally, it is obtained that a Whitney smooth family ofsmall-amplitude quasi-periodic solutions for the equation by developing an abstractKAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional Hamiltonian systems.