Duffing - incoming phase problem

As we all know, Duffing equation go to grate scale period when incoming signal fit to some criterion (sinusoidal and amplitude of resulting signal must be greater than some constant - one can give constant from Lyapunov exponent method). In theory all seems great...
But in practice we receive signal with unknown phase! Look at the picture.

phase (btw how can I convert phase to radians on picture???) have great impact on energy of the output signal. It's simple to understand, on some phases incoming signal suppress internal signal of the Duffing, depends on internal signal. For example our incoming phase is pi then => Acos(wt) + A1cos(wt + pi) = (A-A1)cos(wt). But we want to increase internal energy (amplitude) A.

This is the problem of Duffing. Need to solve it.