Identification of topological information from orbits in area-preserving maps.

We have developed a numerical method to identify invariant-circle families in island-around-island hierarchies and distinguish island regions (including sticky layers) from the surrounding chaotic sea in area-preserving maps. The method uses a rotation-angle representation constructed with delay coordinates, estimates rotation numbers with an evaluated error bound, determines resonance mode numbers of island chains, and classifies orbit types through sparse sampling of the waveform of the rotation angle. Validity tests were performed using the standard map, with benchmarks against widely used methods. We also present a practical application to the magnetic field in the toroidal plasma confinement device Heliotron J. We found that the method achieves accurate extraction of topological information, despite its limitations in certain cases.