Trajectory Dispersion: A Geometric Observable of Pathological State Transitions in Dynamical Biological Systems

Pathological dynamical transitions may emerge not primarily through shifts in population means, but through gradual loss of dynamical coherence characterized by: - increased variability, - weakened attractor organization, - expanding trajectory geometry, - and growing dynamical heterogeneity. This work introduces Trajectory Dispersion (TD), a geometry-based dynamical observable for quantifying trajectory-ensemble organization in biological systems. Using negative controls, synthetic oscillatory systems, progressive attractor destabilization, and experimental Drosophila behavioral trajectories, the study demonstrates that TD captures statistically robust and biologically interpretable signatures of dynamical regime organization inaccessible to static variance-based observables. Rather than classifying pathology directly, TD measures the evolving geometry of biological dynamics and provides a framework for studying pathological transitions as progressive loss of coherent state-space organization. Includes: - full methodological framework, - reproducible Python implementation, - statistical validation pipeline, - supplementary analyses, - and complete reproducibility appendices. Author: Josef Piskac (2026) License: CC BY-NC-ND 4.0 International License