Linearly Stable but Nonlinearly Unstable: Resonances, Krein Signatures, and the Gap Between Spectral and Lyapunov Stability in Hamiltonian Systems

This paper examines the precise relationship between linear stability and Lyapunov stability for equilibria of analytic Hamiltonian systems. Linear stability means that the linearized equations have only bounded solutions, typically because all eigenvalues lie on the imaginary axis and the linearization is diagonalizable. Lyapunov stability is the stronger requirement that solutions starting sufficiently close to the equilibrium remain close for all future time. The central observation is that linear stability does not imply Lyapunov stability. When an elliptic–elliptic equilibrium possesses a low-order resonance of sum type whose resonant coefficient is nonzero, nonlinear terms can drive the actions to infinity even though the linearization itself is stable. Cherry’s classical Hamiltonian provides an explicit illustration: two oscillators in exact two-to-one resonance yield only closed orbits at linear order, yet the cubic resonant term produces solutions that start arbitrarily near the origin and escape to infinity in finite time. The same phenomenon appears in the planar circular restricted three-body problem at the triangular Lagrange points. Below the Routh critical mass ratio the equilibrium is linearly stable, yet at the specific resonant mass ratios corresponding to the two-to-one and three-to-one frequency relations it is Lyapunov-unstable. In both cases the Krein signatures of the modes determine whether a given frequency relation is of sum type (dangerous) or difference type (harmless and confining). The converse implication is more rigid. If the linearization possesses an eigenvalue with positive real part, then the equilibrium is Lyapunov-unstable for any continuously differentiable vector field; the linear divergence dominates near the origin. A borderline exception occurs when the spectrum is pure imaginary but the linearization is non-semisimple, so that linear solutions grow secularly. At the Routh value itself the triangular points furnish such an example: the linear flow is unbounded, yet the nonlinear equilibrium remains Lyapunov-stable because it is enclosed by invariant tori. All of the results are local and finite-order. They concern a neighbourhood of the equilibrium whose size is limited by the distance to the nearest dangerous resonance, and they do not extend automatically to systems with three or more degrees of freedom, strong dissipation, or global phase-space structures. Within these limits, the arithmetic of the frequency vector together with the Krein signatures completely classifies the local possibilities.