Consider a discrete-time dynamical system of the following kind. Assume $x(t) \in \mathbb{R}^n$.
$x(t+1) = A_{\Omega(x(t))} x(t)\quad$ where $\quad A_i = \left [ \begin{array}{c|c} 1 & 0\\ \hline b_i & B_i \end{array} \right ] \quad$ where $\quad \exists (\| \cdot \|_\triangle). \forall i. \| B_i \|_\triangle < 1$.
Here, $\Omega: \mathbb{R}^n \rightarrow \{1,2,\dots,K\}$ has the form $\Omega(x) = \operatorname{argmax}_i 1^\top A_{i} x$, where we denote by $1$ the vector of all ones.
The vector $b_i$ is arbitrary. The matrices $B_i$ are also arbitrary, but they are a contraction with respect to some vector norm (i.e. the corresponding operator norm is less then one). The norm is the same for all $B_i$s.
The task is to describe the limiting trajectories of the system.
My thoughts are:
The function $\Omega$ is a partition of the state space into sets each of which is an intersection of a finite number of half-spaces and therefore convex.
It is easy to see that the system (even for an arbitrary rule $\Omega$) will never diverge in the sense that the trajectory is always contained in some ball around the origin which depends only on the initial condition but not on $t$, since the eigenvalues of any product of such matrices consist of single eigenvalue one and a set of eigenvalues within the complex unit circle.
- Numerical experiments seem to suggest that the system converges to a periodic orbit; but is it always the case? Does it converge to one cycle (a periodic orbit)? Is the cycle finite in the sense that the sequence $\Omega(x(t)),\Omega(x(t+1)),\Omega(x(t+2)), \dots$ eventually becomes exactly periodic? Is the limiting cycle always the same cycle (when we treat two cycles as identical if they produce identical sequence of functions $\Omega(x(t)),\Omega(x(t+1)),\Omega(x(t+2)), \dots$)? If there are many such cycles, is there a way of characterizing the set of such cycles?
- I have done some search and found two approaches: the common Lyapunov function approach, which as far as I know can only be used to prove a system converges to zero (which this system clearly doesn't) and the impact map approach, which, as far as I understand it, doesn't easily extend to discrete-time systems.
I would be grateful even for partial answers or pointers to literature.
Maybe it is possible to exploit the special structure of this system to analyze it with elementary tools?