As a warm up before analyzing the driven damped anharmonic oscillator, we’ll explore the oscillations of a particle in an anharmonic potential
V(x) = 1/2ω₀2x2 + 1/3αx3 + 1/4𝛽x4,
with no damping force and no driving force. Landau uses perturbation theory to predict that for oscillations of amplitude a, the frequency is approximately ω = ω₀ + Δω,
where Δω = (3𝛽/8ω₀ - 2/12ω₀3)a2 = κa2 But just how good is this formula? We’ll find out.



  First set α = 𝛽 = 0 and check that the frequency doesn’t depend on amplitude.

  Next, set α = 0, 𝛽 = 1 and try amplitudes (initial x, taking initial v = 0) 0.1, 0.5, 1, for example, see how the frequency changes. Can you give a qualitative explanation?

  Next, set 𝛽 = 0 and take gradually increasing values of α. At some point, things will go bad. Why?

  Explore possibilities with both α,𝛽 nonzero.

  Here's the relevant lecture.

  Program by: Carter Hedinger