Why does jiggling a ball in a nonlinear potential well lead to multiple harmonics?

Question

Here is my problem: In MATLAB, I am placing a ball inside of a potential energy well, I have control over what this potential energy well looks like (a parabola for example). I then drive the ball with a force function of my choice (sin(t) for example). I then look at the resulting motion of the ball over time. This is something that I can take the Fourier Transform of to determine the frequency content of the ball's motion within the well.

When I place the ball in the bottom of a parabolic potential well and drive it with sin(t), the FT reveals two peaks: one at the frequency I'm driving it, and one frequency that corresponds to the steepness of my parabola (the resonant frequency of my parabola). Even when I drive the ball with a sum of sine functions with different frequencies, I get n+1 peaks in my FT (if n is the number of distinct frequency sine functions I'm driving with). There is no harmonics and no sum frequency or difference frequency generation.

When I place the ball in the bottom of an asymmetric potential well (x^2/(x+1)) and I drive the ball with sin(t) I get all the harmonics of sin(t), all the harmonics of the resonant frequency of the potential, and all the sum and difference frequencies that can be made.

My question is: Why does jiggling a ball in an asymmetric well lead to the generation of sum and difference frequencies? I'm aware they fall out of the formula sin(a)sin(b)=1/2(cos(a-b)-cos(a+b)), but it's still unclear to me why an asymmetric potential would cause two sine waves to be 'multiplied'. I'm not looking for an explanation where the frequencies fall out of some formula, I'm really looking for a fundamental/'intuitive' explanation.

Answer 1

Unless you are very careful about the initial conditions, the system response will be the sum of two components: the "forced response" at the excitation frequency, and the "free vibration" response if you give the ball an initial nonzer position and/or velocity, but there is no external forcing.

Since there is no damping in your system, both responses continue indefinitely.

In the first case, you have a linear system, each component of the response is a sinusoidal vibration, and the two components do not interact with each other. Therefore your Fourier decomposition has two peaks, at the forcing frequency and the free vibration frequency.

If the external force is the sum of $n$ sinusoidal terms, the response will be the superposition of the "forced" response to each one individually, plus the "free" response of the system, giving $n+1$ peaks in the Fourier decomposition.

In the second case, the system is nonlinear, and the two components of the response are both periodic but not sinusoidal. Therefore you see harmonics of their two fundamental frequencies in the Fourier decomposition. Because the system is nonlinear you also get interaction between the two components which gives you all the sum and difference frequencies.

You might want to find some literature about the Duffing equation, which models a simple nonlinear oscillator and has a parameter that specifies the amount of nonlinearity.