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Fig. 1.

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Vertical action and its evolution. Left and middle columns: 2D maps of the vertical action Jz plotted in different coordinates, (Lz, JR′=JR − θ(−Lz)Lz) (first column), (Lz, ωz/κ) (second column), and (2(Ω − Ωp)/κ, ωz/κ) (third column). Top row: Average value of the action in the pixel. Bottom row: Mean changes in Jz from t = 300 to t = 400 (actions and frequencies refer to the moment t = 400). The vector fields in the bottom rows demonstrate the average flow of orbits in given planes over five time units (from 400 to 405). The thin gray lines in each plot represent isodensity contours. The thick gray lines in the top left plot correspond to the lines with the same value of the adiabatic invariant Jv = JR′+Jz + Lz/2. The vertical green lines in the middle column indicate the location of the CR and ILR. Right column: Snapshots (xy and xz) of the bar orbits (abnormal orbits; see Zozulia et al. 2024) at t = 400 with different frequency ratios ωz/κ > 1 (top two plots) and ωz/κ < 1 (bottom two plots).

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