2026/07/19 by Huining Xue
#math.DS
We consider x+f(x,ωt) x+g(x,ωt)=0, where f(x,θ)=∑j=0maj(θ)x2j+1, g(x,θ)=x2n+1+∑j=0n-1bj(θ)x2j+1. The coefficient functions are real analytic and even on the torus and the frequency vector ω is Diophantine. If n≥2(m+1), we construct codimension-one reversible KAM tori accumulating at infinity and prove that all solutions are bounded. The main point is a finite normal-form procedure. After the reversible polynomial reduction, a logarithmic Fourier cut-off is introduced. At the v-th step a truncated homological equation is solved on a non-resonant action interval and the new error satisfies an explicit finite-step recurrence. Thus an arbitrarily small negative power of the large action is reached after finitely many steps. Finally, the Largrangian stability and the existence of quasi-periodic solutions are proved by the reversible KAM theorem.