I have the following system of differential equations:
$\dot{x}_1=f_1\left(\dot{x}_1,x_1,x_2\right)\\
\dot{x}_2=f_2\left(\dot{x}_1,x_1,x_2\right)$
where $\dot{x}_1$ and $\dot{x}_2$ are time derivatives of $x_1$ and $x_2$, respectively. The functions $f_1$, $f_2$ are differentiable, but strongly nonlinear. Thus, I cannot express $\dot{x}_1$ and $\dot{x}_2$ on the left hand side as functions of only the state space variables $x_1$ and $x_2$.
Is there a way to transform the above equations into proper state-space form?
Best Answer
Maybe not the answer you are looking for. To get a more specific answer, more information about the function $f_i$ are required. In some sense the question is as hard as finding an explicit form for an implicit equation.
Theoretical local state-space form
If you consider the function $$ g_i(y_i,x_1,x_2) := y_i - f_i(y_i,x_1,x_2) $$ and let $\widetilde p = (\widetilde y_1, \widetilde x_1, \widetilde x_2)$ be a point with $g(\widetilde p) = 0$. Then, an application of the implicit function theorem yields a function $\phi_i(x_1,x_2)$ locally defined around $\widetilde p$, such that for all $(y_i,x_1,x_2)$ close to $\widetilde p$ we have $$ y_i = \phi_i(x_1,x_2) \iff y_i - f_i(y_i,x_1,x_2) = 0. $$
Therefore, a local state space form is given by $\dot x_i = \phi_i(x_1,x_2)$.
The requirements on $f_i$ are, that $\frac{\partial f_i}{\partial y_i}(\widetilde p) \neq 0$ and that $f_i$ is (at least) continuously differentiable.
Calculus
The derivatives of $\phi_i$ can be expressed in terms of the derivatives of $f_i$, therefore this approach also allows you to carry out some local analysis.
Theoretical global state-space form
This is in general not possible. Take $$f(\dot x,x) = (\dot x-1)(\dot x+1),$$ obviously, the local state-space forms are either $$ \dot x = 1 \quad \text{or} \quad \dot x = -1. $$ But none of these are global and it depends on the initial conditions which track you follow. The situation can be arbitrary complex.
Therefore, I would doubt that a general trick exists, which transforms you system into an explicit equation without strict assumptions on $f_i$.
Numerics
Depending on our task, it might be sufficient to use some root-finding algorithm to get the value of $\phi_i$. (But you need to provide initial guesses for $\dot x_i$, for example the previously computed value of $\phi_i$.)
But, I'm not too familiar with numerics for implicit ODEs.