Piecewise Recursive Sequences with Adaptive Thresholds : Boundary Convergence and Applications

We study discrete-time dynamical systems that switch between different evolution rules based on thresholds that themselves adapt over time. Specifically, we analyze the coupled recursion $a_{n+1} = f(a_n)$ if $a_n \leq c_n$ and $a_{n+1} = g(a_n)$ if $a_n > c_n$, where the threshold evolves according to $c_{n+1} = h(a_n, c_n)$. By transforming the system into triangular coordinates, we map the problem to a piecewise-smooth system with a fixed switching boundary. We derive explicit local stability criteria based on the lower-triangular structure of the associated Jacobians and establish a ``common-limit constraint'': we prove that any convergent orbit that switches regimes infinitely often must converge to a limit where the state and threshold coincide. To demonstrate the framework's utility, we develop an asset-pricing model where investor sentiment follows a ``trailing-stop'' rule. We characterize the parameter regions for market stability versus bubble formation and numerically extend the analysis to coupled markets, illustrating how adaptive thresholds can facilitate the propagation of instability and contagion across financial networks.