This work introduces the Ectochronic Power Operator (ÊP), a projection-based framework for extracting coherent structures from noisy dynamical systems. A formal distinction is established between the operator ÊP, which acts on the signal power, and the scalar metric EP, which quantifies the proportion of structural energy retained after transformation. The method is validated on synthetic datasets in both one-dimensional and two-dimensional regimes. In 1D, the operator successfully recovers soliton-like structures and identifies a natural scale through maximization of EP, achieving high event contrast and low reconstruction error. In 2D, the operator reveals limitations in multiscale fields, where energy-based selection alone collapses to small-scale features. To address this, a topological correction is introduced through the coherence score: Scoh = (EP * Rₑvent) / (KP * Nclusters) This formulation incorporates structural energy, event contrast, noise contribution, and spatial fragmentation, enabling meaningful dominant-scale selection in complex systems. The results demonstrate that while EP is sufficient for single-scale dynamics, multiscale systems require topological weighting to identify physically relevant structures. This work provides a general, model-independent framework for coherent structure detection, with potential applications in nonlinear dynamics, turbulence, signal processing, and spatially extended systems.
Luis Francisco Ramirez Olguin (Tue,) studied this question.