If we take a harmonic oscillator and add an x4 term, to give a potential for example 12x2 + 14x4, the frequency of oscillation of a particle will approach that of the harmonic case as the amplitude of oscillation becomes small, but for amplitudes of order one and more the steeper sides of the second term dominate, and the frequency becomes more rapid.

Therefore, if we want to drive the oscillator to a large amplitude oscillation, beginning at rest, we start driving in resonance with the harmonic oscillator, but as the amplitude increases we increase the driving frequency slowly and the amplitude gradually increases.

But then there’s a surprise: At a certain frequency, depending on the strength of the driving force, on slightly increasing the frequency (keeping everything else constant) the amplitude drops dramatically, typically by a factor of two or so, and stays there, in fact decreasing for any further increase in frequency.

This unexpected phenomenon is explained in Landau’s book, and reproduced in my online notes. You can explore it yourself using this applet.



  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 particle   to 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, f = 0.5. (Or just reload - these are the default values.)

   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