I have a fairly large set of nonlinear equations (16 equations):
$f(x_1,x_2,x_3,\ldots, x_{16}, \theta_1,\theta_2, \ldots, \theta_q)=0$
Where $\theta_i$ are parameters. I would like to find the reference system shift, i.e. $x_i \rightarrow \hat{x_i}+x_{i_c}( \theta_1,\theta_2, \ldots, \theta_q)$ (where $x_{i_c}( \theta_1,\theta_2, \ldots, \theta_q)$ is the shift), so that:
$\hat{f}(0,0,0,\ldots, 0, \theta_1,\theta_2, \ldots, \theta_q)=0$
In other words I would like to shift the coordinate system of the following dynamical system:
$\dot{x}=f(x,\theta)$
So that the equilibrium point of this system:
$\dot{\hat{x}}=\hat{f}(\hat{x},\theta)=f(\hat{x}+x_c(\theta),\theta)-f(x_c(\theta),\theta)$
lays in the origin. How can I manage to do prove that such a transformation exists?