Sharkovskii-type conjecture for bifurcations on manifolds

Let $M$ be a smooth, orientable, metrizable manifold of dimension at least $2$. A bifurcation ordering conjecture. There exists a class of bifurcations $\{B_\alpha\}_\alpha$ and a partial dynamical ordering on $\{B_\alpha\}_\alpha$, both depending on $M$, such that if a $C^1$ one-parameter family $\dot{s}=F_\tau(s)$, $\tau\in I$, undergoes the bifurcation $B_\alpha$ at an interior parameter $\tau_\alpha\in I$, then for every $B_\beta<B_\alpha$ with respect to the dynamical ordering, there exists at least one parameter $\tau_\beta$ interior to $I$ at which $\dot{s}=F_\tau(s)$ undergoes the bifurcation $B_\beta$. This proposes a Sharkovskii-type dynamical ordering for bifurcations on arbitrary smooth manifolds of dimension at least $2$; the existence of the relevant ordering and its asserted forcing property are presented as extensions of the paper's study of topology and bifurcations via the Entropy flow.