A property of an equilibrium (or invariant set) of a dynamical system whereby, for every neighborhood size of the equilibrium there exists a possibly smaller neighborhood of initial conditions such that all trajectories starting in that smaller neighborhood remain within the prescribed neighborhood for all future time.

Principle

Small perturbations to initial conditions result only in uniformly small deviations of trajectories for all forward time; the equilibrium resists finite perturbations in the sense of boundedness, not necessarily convergence.

Demonstration

A damped mechanical oscillator with linear viscous damping returns trajectories that, if initialized sufficiently close to the rest position, never leave a small preset amplitude bound for all future time — hence the rest is Lyapunov stable (though not necessarily asymptotically stable without dissipation).

Misapplication

Equating Lyapunov stability with asymptotic or exponential stability (convergence to the equilibrium) or using a linear test without validating that the linearization captures the nonlinear behavior in the neighborhood of interest.

Consequence

Establishing Lyapunov stability gives control over long-time boundedness of perturbations and is a prerequisite for stronger notions of stability used in control design and robustness analysis.

Reversal

Instability: there exists some arbitrarily small perturbation that produces trajectories leaving any preassigned neighborhood of the equilibrium at some future time, indicating sensitivity and possible divergence.

Boundary

Concerns deterministic dynamical systems and topological neighborhoods in state space; does not by itself describe rates of convergence, finite-time behavior, or probabilistic/stochastic stability notions without additional structure.

Semantic Tension

Close to structural stability (persistence of qualitative behavior under perturbations of the system): Lyapunov stability refers to robustness of trajectories near an equilibrium, whereas structural stability refers to robustness of the phase portrait under changes to the vector field.

Synthesis

Lyapunov stability characterizes equilibria whose nearby trajectories remain uniformly bounded within any prescribed neighborhood for all future time, formalizing the intuitive idea that small initial disturbances do not grow unboundedly.