Thin Chaos Inside a Stable Region: Where Confined Chaos Becomes Escape near an Elliptic Equilibrium

Doucette’s fifth paper examines the structure of the region that a stability certificate protects around an elliptic equilibrium. In two degrees of freedom, on the zero-energy section of an equilibrium whose modes carry opposite Krein signatures, chaos generically exists in every neighbourhood. That chaos is nevertheless trapped so long as it remains inside an invariant circle surrounding the equilibrium; the author calls such chaos thin and the outermost such circle the edge. Beyond the edge the chaos is thick and can lead out. Thin chaos is confined for all time by the Jordan-curve property of the circle, while thick chaos requires a negative-energy mode—even though chaos itself does not. The twist of the section map is proportional to Arnold’s determinant, a relation verified numerically across the mass-ratio range of the triangular points; the twist vanishes at the exceptional ratio where the determinant itself vanishes, producing a shearless circle whose location moves toward the equilibrium as the ratio is approached. At the planar triangular point the edge is measured from rotation-number profiles obtained with weighted Birkhoff averages, a diagnostic independent of integration time. Near the order-three resonance the edge scales linearly with the distance to the resonance over a decade, consistent with the companion paper’s truncated predictions; near the order-four resonance the shrinkage is confined to a narrower window. Escape radii defined by departure within a fixed number of returns systematically exceed the edge by large factors and remain unchanged when the integration length is increased sixteenfold, so they measure stickiness as well as the boundary of confinement. For the planar Penning trap, axial symmetry renders the motion integrable and eliminates chaos entirely. A homogeneous cubic imperfection acts by an exact dilation, so the entire thin–thick structure, including the edge, scales inversely with the imperfection strength—an exact law confirmed numerically. The paper is careful to distinguish what is proved from what is measured, and to note that the edge is resolved only along a single ray and only to the spacing of the profiles. Together with the four companion papers it completes a structural claim for the Trojan Universality Class: every exit mechanism examined in the series requires a negative-energy mode for departure from a neighbourhood of the equilibrium, while the mere presence of chaos does not.