Is it possible to transform a cubic ODE to a set of quadratic ODEs? I am wondering if it is possible (or why it's not possible) to transform a set of cubic ODEs into a larger set of quadratic ODEs. Take as a simple example the ODE $$\dot{x}_1 = x_1^3$$ then can one rewrite this ODE in the form 
\begin{align}
\dot{x}_1 =& f_1(\mathbf{x})\\
\vdots\\
\dot{x}_n =& f_n(\mathbf{x})\\
\end{align}
where each $f_i(\mathbf{x})$ is, at most, quadratic in the variables $\mathbf{x} = (x_1,...,x_k)$? 
A more general question is then: is it possible (and does there exist a general method) to transform a cubic ODE in $n$ dimensions to a quadratic ODE in $m>n$ dimensions? 
 A: Yes. Given the system of cubic equations
$$\dot x_i=\sum_{j,k,l}a_{ijkl}x_jx_kx_l+\sum_{j,k}b_{ijk}x_jx_k+\sum_jc_{ij}x_j+d_i,\quad1\leq i\leq n$$
we can define new variables $y_{ij}=x_ix_j$ to make them quadratic:
$$\dot x_i=\sum_{j,k,l}a_{ijkl}x_jy_{kl}+\sum_{j,k}b_{ijk}y_{jk}+\sum_jc_{ij}x_j+d_i,\quad1\leq i\leq n$$
and introduce new equations for the new variables, which can also be made quadratic:
$$\dot y_{ij}=\dot x_ix_j+x_i\dot x_j$$
$$=\left(\sum_{m,k,l}a_{imkl}x_my_{kl}+\sum_{m,k}b_{imk}y_{mk}+\sum_mc_{im}x_m+d_i\right)x_j+x_i\left(\sum_{m,k,l}a_{jmkl}x_my_{kl}+\sum_{m,k}b_{jmk}y_{mk}+\sum_mc_{jm}x_m+d_j\right)$$
$$=\left(\sum_{m,k,l}a_{imkl}(x_mx_j)y_{kl}+\sum_{m,k}b_{imk}(y_{mk}x_j)+\sum_mc_{im}(x_mx_j)+d_ix_j\right)+\left(\sum_{m,k,l}a_{jmkl}(x_ix_m)y_{kl}+\sum_{m,k}b_{jmk}(x_iy_{mk})+\sum_mc_{jm}(x_ix_m)+x_id_j\right)$$
$$=\left(\sum_{m,k,l}a_{imkl}y_{mj}y_{kl}+\sum_{m,k}b_{imk}y_{mk}x_j+\sum_mc_{im}y_{mj}+d_ix_j\right)+\left(\sum_{m,k,l}a_{jmkl}y_{im}y_{kl}+\sum_{m,k}b_{jmk}x_iy_{mk}+\sum_mc_{jm}y_{im}+d_jx_i\right),$$
$$1\leq i\leq j\leq n.$$
There are $\tfrac{n(n+1)}{2}$ many $y$'s, and $n$ many $x$'s, so the total dimension is $m=\tfrac{n(n+3)}{2}$.
