Researchers have developed a new technique, called Sliding-Window Frequency Ratio (SWFR), to identify and analyze transient resonances in chaotic orbits within relativistic spacetimes. Traditionally, long-term Poincaré sections tend to blur chaotic orbits, obscuring the sequence in which an orbit explores its local structures. The SWFR method overcomes this limitation by reconstructing the temporal order of the orbit, revealing periods where chaotic motion temporarily approximates resonant behavior.

The SWFR technique is constructed solely from radial and polar turning events of the orbit. By analyzing these events within sliding windows of a fixed number of radial cycles, the method counts polar events while keeping track of event times. This allows for the extraction and analysis of any candidate interval as its own Poincaré section. Unlike other methods, SWFR requires no spectra, reference centers, or basis functions, simplifying its application. In tests with integrable Kerr benchmarks, the technique successfully recovered known frequency ratios, and for regular motion, the counting error bound falls off as O(W⁻¹), where W is the window size.

The primary application of SWFR lies in the analysis of chaotic orbits. For instance, in magnetized Kerr spacetime, a charged orbit was observed to dwell near 3/5 and later near 4/7 frequency ratios, with corresponding Poincaré sections exhibiting fivefold and sevenfold patterns. Another orbit showed a temporary residence near 1/2 with two lobes. In Schwarzschild-Melvin spacetime, a photon held a 4/5 ratio across a fivefold pattern. In every case, phase-space geometry and the integer counts from SWFR agreed that globally chaotic orbits are temporarily visiting resonances, opening new avenues for understanding complex dynamics in strong gravitational fields.