Lagrange Stability for Reversible Duffing Equations with Quasi-Periodic Coefficients

Abstract

We consider \[ \ddot x+f(x,\omega t)\dot x+g(x,\omega t)=0, \] where \[ f(x,\theta)=\sum_{j=0}^{m}a_j(\theta)x^{2j+1},\qquad g(x,\theta)=x^{2n+1}+\sum_{j=0}^{n-1}b_j(\theta)x^{2j+1}. \] The coefficient functions are real analytic and even on the torus and the frequency vector $\omega$ is Diophantine. If $n\geq2(m+1)$, 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.