On the formation of the 1:2 resonance in oscillator dynamics

The dynamics of nonlinear oscillators are investigated. We study the formation of $1:2$ resonance in nonlinear periodically forced oscillators due to period doubling of the primary $1:1$ resonance, or born independently. We compute the amplitude-frequency implicit function, the steady-state asymptotic solution, for the effective equation approximating coupled oscillators. Working in the framework of differential properties of implicit functions, we demonstrate that birth of $1:2$ resonances corresponds to singular isolated points of the implicit functions. We provide numerical examples illustrating our theoretical findings.