Consider some function $f(x,p):\mathbb{R}^6 \times \mathcal{D} \subset\mathbb{R}^n \to \mathbb{R}^6$ which transforms vectors $x$ in dependence of parameters $p$. The evolution of $x$ is given by
$ z = \begin{bmatrix} f(x, p) \\ f(f(x,p),p) \\ f(f(f(x,p),p),p) \\ ... \end{bmatrix} $
and thus is given by repeated application of $f$. I'm interested in fitting some observation $\hat{z}$, i.e. find $p^*$ such that
$ || z(p^*) - \hat{z} ||_2 \leq || z(p) - \hat{z} ||_2 \quad \forall p \in \mathcal{D} $
is fulfilled. The function $f$ is differentiable, and the problem has a unique minimum in case the evolution $z=[f(x,p)]$ consists of a single element. However, the problem becomes more complicated in case of longer evolutions $z$, as local minima appear.
Albeit this general form of the problem I have a hard time finding similar optimization tasks, and I would appreciate a keyword describing this particular situation enabling further research. Presumably, this type of problem is encountered frequently when dealing with periodic phenomena?