Classification in Active Dimension 2 for Weighted Residual Dynamics

TL;DR

This paper analyzes weighted residual dynamics in finite-dimensional systems with a focus on active dimension two, classifying the behavior of nonlinear recursions and describing their limits, revealing thresholds for higher-dimensional complexity.

Contribution

It provides a complete classification of nonlinear recursions in active dimension two for weighted residual dynamics, and identifies thresholds for higher-dimensional behavior.

Findings

Recursion can be classified as either stable or collapsing in active dimension two.

The limit behavior of the system can be explicitly described in the two-dimensional case.

A threshold is identified beyond which higher-dimensional dynamics become more complex.

Abstract

We study weighted residual dynamics associated with a rank-one projection in finite dimension. The iteration reduces, after finitely many steps, to a nonlinear recursion on a stabilized active subspace. We prove that this recursion can be classified when the active dimension is two: either a transverse reducing direction persists unchanged, or the coupled part collapses completely. As a consequence, we obtain a description of the limit in the active two-dimensional case and identify the threshold beyond which higher-dimensional behavior becomes more flexible.