Interior structure of black holes with nonlinear terms

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