A Two-Channel F-Transform Representation for Early Trajectory Characterization in Iterated Correlation Dynamics

Abstract

Many nonlinear iterative systems generate high-dimensional trajectories whose early behavior is informative but difficult to compare directly. This paper derives a fixed-dimensional F-transform coordinate representation for early trajectories of iterated Pearson correlation matrices. The construction is defined on the first five-point post-transient signal window, which is the shortest sampled window that simultaneously places the three symmetric fuzzy nodes at observed positions and supports a nondegenerate centered first-degree F-transform coefficient, thereby providing the earliest feasible local level--trend characterization within this sampled geometry. The representation combines two logarithmic observables of the post-transient dynamics: step size and contraction ratio. Applying this same four-coordinate construction to the step-size and contraction-ratio signals yields the eight-dimensional descriptor $\Psi =(v_1,v_2,v_3,s_2,u_1,u_2,u_3,r_2)$, with a common coordinate form across matrix sizes. For the fixed construction, $\Psi =M(q_2,\dots ,q_7{)}^{\top }$, $q_k=\log {\delta }_k$, with $\mathrm{rank}M=6$. Thus, the descriptor is an injective, overcomplete representation of the six logged step sizes underlying the two channels. The representation is Lipschitz stable, and the centered first-degree coefficient recovers affine trends exactly. Convergence-length approximation is used as a downstream test of retained dynamical information. Across 22 matrix dimensions and 22,000 trajectories, repeated train--test evaluation shows predictive performance comparable to raw two-channel and statistical-summary representations. PCA shows that the first two principal components explain on average $84.47\%$ of the descriptor variance. Clustering reveals reproducible coarse organization, with the strongest mean silhouette at $k=2$ and high stability for smaller numbers of clusters.

33 pages, 4 figures, 7 tables