Boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequency (Q281653)

Consider a scalar differential equation of the form NEWLINE\[NEWLINE \ddot{x}+x^{2m+1}=\sum_{j=0}^{2m}p_j(\omega t)x^j, NEWLINE\]NEWLINE where \(p_j\) are real-analytic functions on the two-dimensional torus with frequency \(\omega=(1,\alpha)\).NEWLINENEWLINEThe authors show that if NEWLINE\[NEWLINE \sup_{n>0}(\log\log q_{n+1})/\log q_n<\infty, NEWLINE\]NEWLINE where \(p_n/q_n\) are the convergents of \(\alpha\), then every solution is bounded.