the poles of a transfer function
Write a transfer function as a fraction H(s) = numerator over denominator. The poles are the values of s that make the denominator zero — the points where H(s) blows up to infinity. They are a small handful of numbers (often complex), and astonishingly, just knowing WHERE they sit in the complex plane tells you how the system rings, whether it settles or runs away, and how fast.
The poles are exactly the roots of the characteristic polynomial — the same roots that, back in the homogeneous theory, produced the natural modes e^(rt). That is no coincidence: a pole at s = r contributes a term behaving like e^(rt) to the response. Read the plane geometrically. A pole's real part sets growth or decay: real part negative means e^(rt) decays (the mode dies out), real part positive means it grows (the mode runs away), real part zero means it neither grows nor dies. A pole's imaginary part sets oscillation: a nonzero imaginary part means the mode wiggles, and its size is the ringing frequency. Complex poles come in conjugate pairs, giving decaying or growing sinusoids.
This single picture ties together stability and natural frequencies. A linear system is stable exactly when every pole lies strictly in the left half-plane (all real parts negative), because then every mode decays. A pole creeping toward the imaginary axis means a barely-damped, long-ringing mode; a pole crossing into the right half-plane means instability. Control engineers design systems by deliberately MOVING the poles — pole placement is steering the whole behaviour by relocating a few numbers.
H(s) = 1/(s^2 + 2s + 5) has poles where s^2 + 2s + 5 = 0, namely s = -1 ± 2i; the real part -1 gives decay like e^(-t), the imaginary part ±2 gives oscillation at frequency 2 — a stable, ringing system.
Left-half-plane poles (real part -1) mean the ringing dies out: the system is stable.
Poles strictly in the left half-plane give stability; a pole ON the imaginary axis is the borderline case — an undamped mode that neither decays nor grows — and usually counts as marginally stable, not stable. Right-half-plane poles mean instability.