Top panel: dynamical map on the parametric plane Ω_bar–M_bar for the L₄-center calculated with the parameters of Table 1 (representing the spiral corotation zone); L₄ is a fixed point only if Ω_bar = Ω_p. We see a wide region of stability (light gray tones) for bar masses of the order of 10⁹M_⊙ and for all values of Ω_bar, except in regions when the spiral corotation zone and the bar’s main resonances overlap. In this case, there appears a wide region of chaos (dark gray and black tones). For bar masses greater than ~10¹⁰M_⊙, L₄ is strongly unstable for any value of Ω_bar (red hatched region). Bottom panel: dynamical spectrum of the L₄ orbit calculated for M_bar = 10⁹M_⊙. The nominal values of the bar’s Lindblad resonances are denoted by vertical dashed lines.