Small divisors problem in dynamical systems and analytic solutions of the Shabat equation (Q961101)

The authors have proved the existence of exact analytic solution of the initial value problem to the Shabat equation \[ f'(z)+q^2f'(qz)+f^2(z)-q^2f^2(qz)=\mu,\quad z\in \mathbb{C},\;f(0)=f_0,\tag{1} \] with two complex parameters \(q\) and \(\mu\) in the case \(|q|=1\) under the Brjuno condition [\textit{A. D. Bryuno}, Tr. Mosk. Mat. Obshch. 25, 119--262 (1971; Zbl 0263.34003)]. In addition they have found a new arithmetical condition which is weaker that the Bruno one and apply it to show the existence of Gevrey-like classes solutions of equation (1).