Perturbation Methods
Perturbation methods build approximate analytical solutions to differential equations by expanding the solution in powers of a small parameter. Multiple-scale (two-timing) expansions treat slow and fast variables as independent, separating a slowly varying average from fast oscillations, and are suited to stiff or periodically forced systems with well separated time scales.
Multiple scales for periodically forced catalytic kinetics (2025)
Colombo Tedesco and co-workers applied a two-time-scale expansion to the coverage equations of dynamic catalysis, with the small parameter defined as the smallest rate-constant amplitude times the forcing period. The leading order gives cycle-averaged coverages from linear algebraic equations (neglecting the covariance between rate constants and coverages), and the first-order correction gives the limit cycle by direct integration with a zero-mean condition. The covariance itself serves as a diagnostic: it grows as the time-scale separation or small-amplitude assumptions fail (Colombo Tedesco et al. 2025). Summary: colombotedesco2025.