Activation of a Third Mode: What Two-Mode Stability Certificates Do and Do Not Guarantee

Doucette’s fourth paper asks what a two-mode stability certificate guarantees when a third mode is present. For elliptic equilibria of Hamiltonian systems with three degrees of freedom, the answer is essentially nothing. An explicit polynomial example is constructed whose restriction to an invariant plane passes every clause of the two-mode admission test, with comfortable margins, and is Lyapunov stable within that plane, yet the full system has solutions that start arbitrarily close to the equilibrium and escape in a time proportional to the initial amplitude. The escaping mode is one of the two observed modes; an observer monitoring only those modes sees growth without any change in the two-mode spectrum. A general criterion is proved for truncated resonant normal forms: if the resonance module contains no direction in which every action can grow simultaneously, a positive combination of the actions is conserved and every orbit remains bounded; otherwise the module necessarily couples modes of both Krein signatures. Explicit Cherry-type and three-wave examples confirm that escape does occur in the latter case. At the spatial triangular Lagrange point, reflection symmetry and the exact vertical frequency of unity yield three precise statements: no resonance of order four or lower involves the vertical mode, every resonance capable of driving growth has high order (at least thirteen at the Sun–Jupiter mass ratio), and fully resonant mass ratios form a dense set of Pythagorean values. A three-mode Birkhoff normal form through order four is computed and validated against integrations; plain quasi-convexity fails everywhere examined, while the isoenergetic determinant does not vanish. For the ideal Penning trap, axial symmetry supplies a second quadratic integral—the energy in the Larmor frame—that is positive definite throughout the trapping region, so every axisymmetric anharmonicity leaves the equilibrium Lyapunov stable regardless of resonances. Different angular orders and parities of imperfections open different channels, as integrations at a triple-resonance point confirm. The paper is characteristically precise about logical status and limitations. The boundedness criterion applies only to truncated forms, two-mode certificates are not inherited, and the spatial triangular points still rely on effective rather than Lyapunov stability. The central demonstration is that certificates computed from two modes say nothing about a third, and that the relevant questions in three degrees of freedom concern the directions opened by the resonance module and the existence of integrals beyond the energy.