Interior structure of black holes with nonlinear terms

We investigate the oscillation of the Kasner exponent $p_t$ near critical point of the hairy black holes dual to holographic superfluid and reveal a clear inverse periodicity $f(T_c/(T_c-T))$ in a large region below the critical temperature. We first introduce the fourth-power term with a coefficient $\lambda$ to adjust the oscillatory behavior of the Kasner exponent $p_t$ near the critical point. Importantly, we show that the nonlinear coefficient $\lambda$ provides accurate control of this periodicity: a positive $\lambda$ stretches the region, while a negative $\lambda$ compresses it. By contrast, the influence of another coefficient $\tau$ 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 actively control from the fourth- and sixth-power term into the black hole interior region.