The Resonance Gap and the Shrinking Stability Region: Why Frequency Separation Does Not Protect an Elliptic Equilibrium

Doucette’s third paper dismantles a common heuristic: that an elliptic equilibrium is well protected whenever its two frequencies are well separated. For Hamiltonian systems with two degrees of freedom the relevant quantity is not the separation itself but the resonance gap—the distance of the frequency vector from every low-order resonance. The separation bounds this gap only from above; an exact resonance of order three or four can occur while the frequencies remain far apart. When the modes carry opposite Krein signatures, such a resonance produces escape. When the signatures agree, the positive-definite energy confines the motion regardless of resonance. At exact resonance the truncated normal forms yield explicit escape-time scalings: inversely proportional to the initial amplitude at order three, and to its square at order four under Markeev’s criterion. Near resonance the region of bounded motion shrinks continuously—linearly with the detuning at order three, and with the square root at order four—while the formal statement of stability remains true. These predictions are tested at the triangular Lagrange points. The exceptional mass ratios at which the frequencies form a three-to-one or two-to-one resonance are classically known to be unstable; the paper computes the relevant normal-form coefficients (validated against the closed form of Arnold’s determinant), confirms that Markeev’s inequality is satisfied by a wide margin, and measures escape times and stable radii in the full planar restricted problem. The observed exponents match the truncated predictions. A selection rule for symmetry-breaking imperfections is also derived for an idealized Penning trap near a three-to-one radial resonance. The paper is careful to separate what is classical from what is new: the explicit dichotomy on signatures, the bounded-region scalings with inner bounds, the full-problem measurements, and a corrected admission test for the Trojan Universality Class that replaces the separation by the resonance gap. Limitations are stated plainly—the analytic results concern truncated systems on an invariant set, the numerical radii depend on finite times and initial conditions, and everything is planar. The central message is that frequency separation alone is not protective, and that proximity to a low-order resonance degrades the domain of stability long before the resonance is reached.