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.
2026-03-24 20:32:01.1774384321
Why does the trace of the Jacobian and its eigenvalues determine equilibrium stability?
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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.