[Math] Why does the trace of the Jacobian and its eigenvalues determine equilibrium stability

Question

I am a high school student studying differential equations!
I just cannot understand exactly why the trace of the Jacobian Matrix and its eigenvalues determine equilibrium stability (I encountered the Jacobian when learning about the Lotka-Volterra equations). Could anybody offer an explanation without invoking topology or Lyapunov stability? Thank you.

Answer 1

A system is stable if all eigenvalues $\lambda _i$, satisfy $$ Re(\lambda _i) < 0.$$ Otherwise the positive real part of eigenvalue will generate an exponential function which diverges and cause the equilibrium to be unstable.