File:Duffing response jumps.svg

English: Phase jumps in the frequency response of the Duffing equation, ${\displaystyle {\ddot {x}}+\delta {\dot {x}}+\alpha x+\beta x^{3}=\gamma \cos(\omega t)\,}$ for ${\displaystyle \alpha =\gamma =1,}$ nonlinear stiffness ${\displaystyle \beta =0.04}$ and damping ${\displaystyle \delta =0.1.}$ The graph shows the amplitude ${\displaystyle z}$ of the steady-state periodic response ${\displaystyle x(t)=z\cos(\omega t-\theta )+\dots }$ as a function of angular frequency ${\displaystyle \omega .}$ The dashed parts of the frequency response are unstable, i.e. at the corresponding forcing frequency the realized response is on either one of the two amplitudes in the drawn-lines part of the graph.
When the frequency ${\displaystyle \omega }$ is slowly varying, the response amplitude ${\displaystyle z}$ can exhibit jumps. When the frequency is slowly increased from very low to high, the amplitude jumps down from A to B. But when the frequency is decreased from high to low, the amplitude jumps at C up to D.
The frequency response is obtained through the method of harmonic balance as:

${\displaystyle \left[\left(\omega ^{2}-\alpha -{\frac {3}{4}}\beta z^{2}\right)^{2}+\left(\delta \omega \right)^{2}\right]\,z^{2}=\gamma ^{2}.}$