Bounce, Turnaround, and the Anisotropy Problem in Cyclic Cosmology on a Brane with a Timelike Extra Dimension
We study cosmological bounces, turnarounds, and cyclic evolution on an anisotropic Bianchi-I brane embedded in a five-dimensional bulk with a \emph{timelike} extra dimension, within the Shtanov--Sahni braneworld framework. Restricting to the flat, dark-radiation-free, effective-$\Lambda$-free branch of the general anisotropic brane Friedmann equation, we drive the dynamics with a single canonical scalar field obeying the uniform-rate condition $\dot\phi=-\lambda=\mathrm{const}$, with shear anisotropy encoded through a geometric term $\Omega_\sigma(a)\propto a^{-6}$. We derive a general turning-point classification valid for any fluid obeying the null energy condition: turnarounds at negative energy density occur unconditionally, while bounces at $\rho>\rho_c$ occur only when the negative high-energy brane correction dominates the decelerating shear term. Specializing to the uniform-rate scalar, we obtain closed-form bounce and turnaround conditions, the leading-order excess of the bounce density above critical, and a matching condition for a finite cyclic branch connecting a bounce at $N_B$ to a turnaround at $N_T$. We identify post-bounce superinflationary and post-shear-dilution ordinary-inflationary regimes, compute the single-field curvature power spectrum, and derive parameter relations fixing $H_*$, $\lambda$, $\rho_c$, and the shear amplitude $\Sigma_g^2$ in terms of the observed amplitude $A_s$ and tilt $n_{s*}$. An explicit CMB-normalized parameter point shows that sustaining a long, weak-shear cyclic phase compatible with observations requires the shear amplitude suppressed by $10^{2}$--$10^{3}$ orders of magnitude below $H_*^2$, depending sensitively on the pivot density fraction $x_*=\rho_{\phi*}/\rho_c$. We discuss the physical origin of this anisotropy problem, its parametric dependence, and the status of the periodicity condition required for a genuinely cyclic $V(\phi)$.