Let $x \in \Delta_n$, where $\Delta_n = \{u \in \mathbf{R}^n_{\ge 0} \mid \sum u_i = 1\}$ is the probability simplex. Suppose that I have an (autonomous) dynamical system $$ \dot{x} = A(x) \cdot x $$ Where A(x) is an $n \times n$ matrix whose entries depend on $x$. By construction (the details are not relevant to the question), I know that this system converges to a unique fixed point $x^\star$, no matter where it starts.
Furthermore, I also know that for any fixed $\hat{x}$, $A(\hat{x})$ is the generator matrix of an ergodic Markov chain, i.e. the system $\dot{x} = A(\hat{x}) \cdot x$ also converges to a unique fixed point (different from $x^\star$ except of course if $\hat{x} = x^\star$.)
Now suppose we do the following:
- Choose $x^1 \in \Delta_n$ as an arbitrary starting point
- let $x^{(k+1)}$ be such that $A(x^k) \cdot x^{(k+1)} = 0$ (i.e., the fixed point of the linear system $A(x^k)$)
My question is: Does $x_k \to x^\star$ as $k \to \infty$?
Some background
The idea behind this question comes from the interpretation of my dynamical system as some kind of nonlinear Markov chain, whose transition rates depend on the current distribution over states.
Informally, the process I describe is can be described as follows.
- "linearize" the Markov chain around the current distribution $x^k$
- let it "cool down" completely, until it reaches the stationary state $x^{(k+1)}$
- go back to step 1, replace $x^k$ by $x^{(k+1)}$
There is some (slightly confused) idea of time-scale separation going on here.
Intuitively, I would say that the answer to my question is yes, but I am unable to formalize my intuition. Any help is appreciated!