In this letter a new solution for adaptive absolute stabilization problem for SISO systems based on the circle criterion and on a new version of passification with respect to given input and output is proposed. It is shown that the proposed conditions are not only sufficient but also necessary for the existence of a Lyapunov function of the type 'quadratic form of state plus quadratic form of parametric errors' which means that these stabilization conditions cannot be improved in the considered class of Lyapunov functions. The proposed results are illustrated by synchronization of two Chua's circuits. In addition to the simple case where control and nonlinearity are located in the first equation, the complex case where control is located in the second equation is also considered. Based on the new passification result a difficult problem of adaptive stabilization with non-matched nonlinearities is solved.