Abstract We investigate the oscillation of the Kasner exponent pₜ p t near the critical point of hairy black holes dual to holographic superfluid and reveal a clear inverse periodicity f (Tc/ (Tc-T) ) f (T c / (T c - T) ) in a large region below the critical temperature. We first introduce the fourth-power term with a coefficient λ to adjust the oscillatory behavior of the Kasner exponent pₜ p t near the critical point. Importantly, we show that the nonlinear coefficient λ provides accurate control of this periodicity: a positive λ stretches the region, while a negative λ compresses it. By contrast, the influence of another coefficient τ is more concentrated in regions away from the critical point. This work provides a new perspective for understanding the complex dynamical structure inside black holes and extends the active control from the fourth- and sixth-power terms into the black hole interior region.
Zhao et al. (Sat,) studied this question.