Revisiting the shadow of Johannsen-Psaltis black holes

The Johannsen-Psaltis (JP) metric provides a robust framework for testing the ``no-hair theorem'' of astrophysical black holes due to the regular spacetime configuration around JP black holes. Verification of this theorem through electromagnetic spectra often involves analyzing the photon sphere near black holes, which is intrinsically linked to black hole shadows. Investigating the shadow of JP black holes offers an effective approach for assessing the validity of theorem. Since the Hamilton-Jacobi equation in the JP metric does not permit exact variable separation, an approximate analytical approach is employed to calculate the shadow, while the backward ray-tracing numerical method serves as a rigorous alternative. For JP black holes with closed event horizons, the approximate analytical approach reliably reproduces results obtained through the numerical computation. However, significant discrepancies emerge for black holes with nonclosed event horizons. As the deviation parameter ${\ensuremath{\epsilon}}_{3}$ increases, certain regions of the critical curve transition into nonsmooth configurations. Analysis of photon trajectories in these regions reveals chaotic dynamics, which accounts for the failure of the approximate analytical method to accurately describe the shadows of JP black holes with nonclosed event horizons.