Duffing response branches and the limits of harmonic balance

Consider the driven Duffing oscillator \[ \ddot x+2\beta\dot x+\omega_0^2x+\alpha x^3 =f\cos\omega t, \] with \(\beta>0\). Assume a periodic response dominated by its fundamental, \(x(t)\approx A\cos(\omega t-\delta)\), and use \(\cos^3\theta=(3\cos\theta+\cos3\theta)/4\). Perform one-harmonic balance to derive the implicit amplitude equation and phase relation. Recover the undamped, unforced backbone relation and explain how the sign of \(\alpha\) determines hardening or softening. Let \(y=A^2\), \(\Delta=\omega_0^2-\omega^2\), \(q=3\alpha/4\), and \(d=2\beta\omega\). Derive the condition for folds of the algebraic response curve at fixed \(\omega\), and show that the candidate fold values satisfy \[ q y=\frac{-2\Delta\pm\sqrt{\Delta^2-3d^2}}3. \] State the existence and positivity conditions. For \(\omega_0=1\), \(\alpha=1\), \(\beta=0.05\), and \(\omega=1.20\), compute the two fold amplitudes and the corresponding force levels. Explain what three algebraic amplitudes between the force folds do and do not establish about dynamical stability, and identify the approximation discarded by one-harmonic balance.

Balance in-phase and quadrature components relative to \(\cos(\omega t-\delta)\), then square and add.