As a warm up before analyzing the driven damped anharmonic oscillator, we’ll explore the oscillations of a particle in an anharmonic potential

with no damping force and no driving force. Landau uses perturbation theory to predict that for oscillations of amplitude *a*, the frequency is approximately
where

But just how good is this formula? We’ll find out.

**Walk-Through**

First set and check that the frequency doesn’t depend on amplitude: that is, on initial *x*, taking initial *v* = 0.

Next, set *α* = 0, 𝛽 = 5, set speed = 20, and try amplitudes (initial *x*'s)= 0.1. Now go to initial *x* = 1, set speed = 2, and see how
the frequency changes. Can you give a *dimensional* explanation? (Note that changing speed doesn't affect the physics, just the rate at which you see it.)

Next, set 𝛽 = 0 and take gradually increasing values of *α*, keeping the amplitude fixed.
At some point, things will go bad. Why? Does the critical value of *α* depend on amplitude? Explain what's going on.

Now set ω_{0}^{2} = -1, *α* = 0, 𝛽 = 5. This is the potential of the Duffing oscillator. Explore.

Here's the relevant lecture.

*Program by: Carter Hedinger*