Showing a bound exists I was able to derive the following differential equations I have to work with for a function $V$:
$$ \begin{align*}
  dV(x_1,x_2,x_3,x_4) &= \left(x_1^2-1\right)\left[(1-x_1^2-x_2^2-x_3^2-x_4^2)-c_{21}x_2^2+e_{31}x_3^2+e_{41}x_4^2 \right]\\
    &+  \left(x_2^2-1\right)\left[(1-x_1^2-x_2^2-x_3^2-x_4^2)+e_{12}x_1^2-c_{32}x_3^2-c_{42}x_4^2 \right]\\
    &+  \left(x_3^2-1\right)\left[(1-x_1^2-x_2^2-x_3^2-x_4^2)-c_{13}x_1^2+e_{23}x_2^2+e_{43}x_4^2 \right]\\
    &+  \left(x_4^2-1\right)\left[(1-x_1^2-x_2^2-x_3^2-x_4^2)-c_{14}x_1^2+e_{24}x_2^2-c_{34}x_3^2 \right] dt\\
\end{align*}$$
What I would like to do is find constants $c_0, c_1, c_2, c_3, c_4$ so that:
$$dV(x_1,x_2,x_3,x_4) \leq c_0+c_1x_1^2+c_2x_2^2+c_3x_3^2+c_4x_4^2 dt$$
All the parameters $e_{ij}$ and $c_{ij}$ are positive, and each $x_i >0$ as well. Clearly this bound may not exist if the $e_{ij}$ are too large since the equations may explode, so the suitable condition  to prevent this is suppose to be $\Pi e_{ij} < \Pi c_{ij}$. What that should do is somehow create enough push from the negative terms to prevent explosion. However, I have been unable to use that condition and show the bound I'm looking for exists. Please help!
 A: 
Clearly this bound may not exist if the $e_{ij}$ are too large since the equations may explode, so the suitable condition  to prevent this is suppose to be $\Pi e_{ij} < \Pi c_{ij}$. What that should do is somehow create enough push from the negative terms to prevent explosion.

The condition $\Pi e_{ij} < \Pi c_{ij}$ is not sufficient. For instance, put $x_1=x_2=1$ and $x_3=x_4=x$. 
Then when $x$ tends to infinity, $$dV(x_1,x_2,x_3,x_4)=(x^4(e_{42}-c_{34}-4)+O(x^3))dt$$
which is asymptotically greater than $(c_0+c_1+c_2 +(c_3+c_4)x^2)dt$, provided  $e_{42}-c_{34}-4>0$. To assure this condition along with  $\Pi e_{ij} < \Pi c_{ij}$ we can put, for instance, $e_{42}=6$, all others $e_{ij}=1$,  $c_{34}=1$, and all others $c_{ij}=2$.
Update. It seems the following. 
For the convenience we extend the domain of $c_{ij}$ and $e_{ij}$, by putting $c_{ij}=c_{ji}$, $e_{ij}=e_{ji}$, and $c_{ii}=c_{jj}=0$ for all $i$ and $j$. Consider  symmetric matrices $C=\|c_{ij}\|$, $E=\|e_{ij}\|$ and $D=\|2_{ij}\|$. Put $A=(C+D-E)/2$. Then 
$$dV(x_1,x_2,x_3,x_4)=(-4+5|x|^2-x^TAx)dt,$$ 
where $$x^T=(x_1^2, x_2^2, x_3^2, x_4^2)$$  and $$|x|=x_1^2+x_2^2+x_3^2+ x_4^2.$$ 
So the function $dV$ is quadratic bounded by a function of  a form $C_1+C_2(|x|)dt$ iff its quartic part $-x^TAx$ is non-positive for all $x$.  The sufficient conditions for this are non-negativity or a non-negative definiteness of the matrix $A$. The matrix $A$ is non-negative iff each $c_{ij}+2\ge e_{ji}$ for each $i$ and $j$. 
