What is the difference between Lyapunov stability and asymptotic stability?

Lyapunov stability means nearby solutions stay nearby for all time, but they need not approach the equilibrium. Asymptotic stability adds the requirement that nearby solutions actually converge to the equilibrium as time increases.

When does linearization fail to decide stability?

Linearization is conclusive only at hyperbolic equilibria, where the Jacobian has no eigenvalues on the imaginary axis. In the borderline non-hyperbolic case, such as a pure center, the nonlinear terms can determine stability, and a Lyapunov function or center-manifold analysis is needed.

Stability Theory of ODEs

Stability theory studies whether the solutions of a differential equation that start near an equilibrium remain near it or return to it over time.

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Definition

An equilibrium is Lyapunov stable if solutions starting sufficiently close remain arbitrarily close for all later time, and asymptotically stable if in addition they converge to the equilibrium; instability means at least some nearby solutions move away.

Scope

This topic covers the definitions of Lyapunov stability, asymptotic stability, and instability, linearization and the Hartman-Grobman theorem, the direct method of Lyapunov functions, LaSalle's invariance principle, and the classification of equilibria of planar systems as nodes, saddles, foci, and centers.

Core questions

Will small perturbations of an equilibrium grow, persist, or decay?
When does the linearization correctly predict the stability of a nonlinear equilibrium?
How can stability be established without solving the equation explicitly?
How are planar equilibria classified by their local phase portraits?

Key theories

Lyapunov's direct method: If a positive-definite function decreases along trajectories, the equilibrium is stable, and strictly decreasing such a function forces asymptotic stability, all without solving the differential equation.
Linearization and the Hartman-Grobman theorem: Near a hyperbolic equilibrium the nonlinear flow is topologically conjugate to its linearization, so the eigenvalues of the Jacobian determine local stability.
LaSalle's invariance principle: When a Lyapunov function is only non-increasing, trajectories converge to the largest invariant set within the region where its derivative vanishes, extending asymptotic-stability conclusions.

Clinical relevance

Stability analysis underlies control engineering, where it certifies that a designed system returns to its operating point after disturbances, and it explains the persistence of equilibria in ecological, physiological, and economic models.

History

Lyapunov's 1892 dissertation founded the general theory of stability of motion and introduced both linearization and the function-based direct method. Poincare's qualitative analysis of planar systems supplied the geometric picture, and the mid-twentieth century added the Hartman-Grobman theorem and LaSalle's invariance principle.

Key figures

Aleksandr Lyapunov
Henri Poincare
Philip Hartman
Joseph LaSalle

Seminal works

perko2001
khalil2002

Frequently asked questions

What is the difference between Lyapunov stability and asymptotic stability?: Lyapunov stability means nearby solutions stay nearby for all time, but they need not approach the equilibrium. Asymptotic stability adds the requirement that nearby solutions actually converge to the equilibrium as time increases.
When does linearization fail to decide stability?: Linearization is conclusive only at hyperbolic equilibria, where the Jacobian has no eigenvalues on the imaginary axis. In the borderline non-hyperbolic case, such as a pure center, the nonlinear terms can determine stability, and a Lyapunov function or center-manifold analysis is needed.