Lyapunov function

A Lyapunov function is a scalar function defined on the phase space of a dynamical system that decreases monotonically along trajectories and attains its minimum at an equilibrium point. Unlike Lyapunov exponents, which quantify instability through linearization, a Lyapunov function proves stability globally without requiring the system to be close to equilibrium. The existence of a Lyapunov function is sufficient for asymptotic stability but not necessary; conversely, the absence of positive Lyapunov exponents is necessary for stability but not sufficient. The two concepts — Lyapunov function and Lyapunov exponent — are complementary pillars of stability theory: one gives global, nonlinear proofs of order, the other gives local, linear measures of chaos. Lyapunov functions are central to control theory, where they are used to design stabilizing feedback, and to the theory of dissipative systems, where they represent the system's free energy or entropy production.