Lyapunov Stability

Energy-like functions certify stability without solving ODEs

Phase Portrait & Lyapunov Level Sets

System:

V̇ along a trajectory

Theory

V(x) ≥ 0, V(0) = 0
V̇(x) = ∇V·f(x) ≤ 0 → stable
V̇(x) < 0 (x≠0) → asymptotically stable
V(x) = x₁² + x₂² (quadratic Lyapunov)

Lyapunov's direct method (1892): if you can find a positive-definite function V whose time derivative is non-positive along all trajectories, the equilibrium is stable. No need to solve the ODE.


Region of attraction: the set {x: V(x) ≤ c, V̇(x) ≤ 0} is an invariant sublevel set — any trajectory starting inside stays inside. This provides a certificate for the basin of attraction.


For Van der Pol, the quadratic V is not a perfect Lyapunov function — V̇ changes sign, reflecting the limit cycle. The system is not globally stable to zero.