Hey anons. I want to know of different ways of finding the stability of an equilibrium point in a system $\dot{x} = f(x)$.

Textbooks fixate on linearization followed by looking at the eigenvalues, but this almost never works in practice. What are some other ways local stability can be determined? I'm familiar with Lyapunov functions, and I recently read about a researcher finding a topologically conjugate system (that wasn't the linearization) where analysis was possible. Do you know of any ways you can do this? If so, where do you read about them?