It’s easy to check how the x4 term increases the frequency of oscillation of the undamped, undriven oscillator.
Set both damping term and driving term to zero, set the initial position of the particleto 0.1 and initial velocity zero. The graph has a frequency
readout, you’ll see it’s close to the harmonic oscillator. Then successively increase the initial displacement, and see how the frequency increases.
Next, we’ll look for the discontinuous amplitude change.
Set ω₀2 = m = 𝛽 = 1, α = 0, 2λ = 0.34.
Now put f = 0.5. Gradually increase γ from 1. The drop occurs around γ = 1.39.
Try now f = 0.3. The curve is now like a distorted resonance, steep on the high frequency side, but not discontinuous. The drop sets in around
f = 0.4. This agrees pretty well with Landau’s estimate.
Here's the relevant lecture.
Program by: Carter Hedinger