Q:

Predict parameter influence in a system of equations

Suppose I have a generic system of "n" equations in "n" unknowns (possibily non linear)
Let's take "n" = 2 as an example here. Calling "x" and "y" the two unknowns, and calling "h" a parameter of the problem, this will reduce to two equations of the form
f(x,y,h) = 0
For example:
1)   y = 2*x + h*(x^2)
and
2)   y = 4*(x^3) + 2*h*x
Now suppose my goal is to estimate how one of the unknowns (say "y") depends on the parameter ("h"). I could clearly try to solve the system above, but this takes time (especially if the system is nonlinear and there are many unknwns).
So my question is: is there a way to estimate the dependence of a SPECIFIC unknown (say "y") from a SPECIFIC parameter (say "h") WITHOUT having to solve the whole system?

A:

The first tool to use when you have systems of equilibrium equations $f(x,h)=0$ with $x \in \mathbb{R}^N$ and want to understand how $x$ responds to a perturbation of $h$ is the Implicit Function Theorem.  The idea is that if the Jacobian of $f$ with respect to $x$ exists and is of full rank, you can implicitly differentiate (by the chain rule) to get
$$
J_x[f(x(h),h)] \nabla_h x(h) + \nabla_h f(x(h),h) = 0
$$
and solve to get
$$
\nabla_h x(h) = J_x[f(x(h),h)]^{-1} \nabla_h f(x(h),h).
$$
So at a point, you can see how a small change to $h$ perturbs the system when you have an equilibrium equation like $f(x,h)=0$.  The IFT actually implies $x(h)$ is a continuous function in a neighborhood of your particular $h$, but you should be careful not to push this result too far without understanding its limitations further.
Entire books have been written on the subject, including an excellent and comprehensive one by Rockafellar.  It includes topics like non-differentiabiliy of $f(x,h)$ in $x$, how large a set the implicit function $x(h)$ is valid for, and the classical proofs based on the contraction mapping theorem.

