It is rather common to use matrices to represent the relationship of the states of dynamical systems. It is very natural to use matrices because of their ease in analysis. Stability, convergence issues naturally follow from this. But I have a problem with a 'stubborn' state whose representation in a dynamical system I can't think of.
Let the state of a system at time $t$ be given by the vector $\begin{pmatrix}x_t \\ v_t \\ w_t \\ z_t \end{pmatrix}$. $x$ and $v$ are nicely (and neatly related) by a certain equation.So it is no problem writing the relationships down.
$z$, on the other hand, is happily stationary. It is constant for all values of $t$. So it is also not a problem in the matrix representation, as well.
The tricky, stubborn part is $w_t$. What it does is this: $w$ chooses the value of $x$ that is nearest the value of $z$ (from all the previous values so far encountered). (Something like: $\min |x_t - z|$.)
How does one represent a system like this in matrix form? Or should I abandon this technique and go for another one?
Please send your insights.
In general, we don't write dynamical systems using matrices, because the RHS of the derivatives are often non-linear. Non-linear stuff is hard to represent in matrix form.
Non-linear flows can be locally described approximately linearly (e.g. using a Jacobian), so things like stability can be determined using certain theories (but only if fixed points are hyperbolic, that is, none of the real part of the corresponding Jacobian matrix's eigenvalues are zero).
The dynamical system you are describing is discrete time though, am I right?