What is the simplest PDE/ODE/model I can use to understand how nonlinearities can lead to leakage of energy to higher harmonics in an oscillator? I came across this problem in the study of surface waves in an oscillating cylindrical vessel of liquid.
There are various eigenmodes described using Bessel functions, and energy transfer can happen between them through nonlinear effects (usually to higher order modes or modes with a frequency that is close to a multiple of the base frequency).
This is a very complicated system to explore nonlinearity in, so I would like to know if there are simpler systems that could be used as an analogy.
 A: I think a very nice and also relevant example is that of second harmonic generation (SHG). I will summarize it briefly, but you can find many resources online.
Maxwell's wave equation in a medium is (setting $c=1$)
$$\nabla^2 E - \frac{\partial^2 E}{\partial t^2} = \frac{\partial^2 P}{\partial t^2},$$
where P is the polarization of the medium. In linear optics one uses the ansatz $P(\omega) = \chi(\omega)E(\omega)$, where $\chi$ is called the linear response function. More generally, one can expand $P$ in a Mac Laurin series, but here we will truncate at the second order nonlinearity:
$$P = \chi E + 2dE^2.$$
The parameter d is a material property, and by symmetry considerations can only be non-zero if the material does not posses inversion symmetry (just as a side info, but not important here). Inserting this expansion into the wave equation gives
$$\nabla^2 E - n^2\frac{\partial^2 E}{\partial t^2} = -S,$$
where $S = -\frac{\partial^2}{\partial t^2}(2dE^2) =: \frac{\partial^2}{\partial t^2}P_\mathrm{NL}$ is called the source term and $n$ is the refractive index.
This can be solved exactly with coupled wave theory, but let's just look at the problem in the Born approximation to get better intuition. Let's assume our zeroth order solution is just a plane wave: $E_0(r,t) = \cos(\omega t - kr)$. Then the second order part of the polarization is
$$P_\mathrm{NL} = 2dE^2 \approx 2dE_0^2 = d(1 + \cos(2\omega t - 2kr)).$$
If you insert this back into the wave equation, you see easily that this gives rise to an electric field oscillating at $2\omega$. In a photon picture you can think of two photons of frequency $\omega$ coming from the "incoming" beam $E_0$ combining into one photon of frequency $2\omega$ (energy conservation is not violated) of the "outgoing" beam $E_1$, where $E_1$ is the first-order Born approximation. So you excite the system with frequency $\omega$, but the 2nd order nonlinearity of the medium shifts that into the second harmonic.
