Asymmetric Duffing oscillator: metamorphoses of $1:2$ resonance and its interaction with the primary resonance

We investigate the $1: 2$ resonance in the periodically forced asymmetric Duffing oscillator due to the period-doubling of the primary $1: 1$ resonance or forming independently, coexisting with the primary resonance. We compute the steady-state asymptotic solution - the amplitude-frequency implicit function. Working in the differential properties of implicit functions framework, we describe complicated metamorphoses of the $1:2$ resonance and its interaction with the primary resonance.