Response to a first-order resonance. Left panel: maximum eccentricity, e_max, reached by a particle initially on a circular orbit with modified semimajor axis ∆ā/a₀, with m < 0 and ϵ′ < 0. The right (resp. left) branch of the function is given by Eq. (29) (resp. 30). The value of e_max suffers a discontinuity at Δā/a₀ = (3/2)(m − 1)|2ϵ′|^2/3, where e_max jumps from |2ϵ′|^1/3 to |16ϵ′|^1/3, the maximum possible eccentricity, e_peak. Right panel: phase portrait corresponding to the discontinuity, with the homoclinic trajectory going through the origin. The red and blue points correspond to their counterparts shown in the left panel.