Bonus: When Is It Really an Oscillator?
Neuroscientists constantly ask whether the brain oscillates in response to a stimulus. There's no agreed definition — it's a bit of an 'I know it when I see it' problem. Math can help pin it down.
The Stuart–Landau system is a famous nonlinear oscillator (no exact solution — Euler is the only way in). Researchers use it to reason about what to expect experimentally. It has a natural rhythm of 4 Hz here.
Drive it with an input and change the input's frequency. Near 4 Hz it locks on and traces a clean loop; push far from 4 Hz and the oscillation breaks down:
The Stuart–Landau system is a nonlinear oscillator (no exact formula — only Euler can solve it). Drive it near its natural 4 Hz and it locks on, tracing a clean loop in the phase plane. Push the input frequency far from 4 Hz and the oscillation breaks down — a reminder that a system can be an oscillator yet not follow every input. This is how researchers reason about brain rhythms (like EEG α-waves).
The insight (from Doelling & Assaneo, 2021): a system can be a genuine oscillator yet fail to follow an input at the wrong frequency. So if you flash a light at 50 Hz and the visual cortex doesn't track it, that doesn't prove it isn't an oscillator — you might just be using the wrong frequency. This is exactly why EEG rhythms (like α-waves) come and go rather than ticking forever.
🎉 That's Math Mountains complete — from derivatives and integrals all the way to simulating brain rhythms. Everything downstream in NMA (dynamical systems, deep learning, modeling) stands on the calculus you just built.