Anisotropic Transition Accessibility in Dynamical Systems: A Geometric Constraint on Reachability

Transitions in complex dynamical systems are typically interpreted as outcomes of intrinsic dynamics or external forcing. Recent work has established that transitions are structurally constrained and occur only within admissible regions of state space. However, a fundamental question remains unresolved: even when transitions are possible, how are they dynamically accessed? Here we introduce anisotropic transition accessibility as a fundamental property of nonlinear dynamical systems, formalized through the concept of Structured Reachability. We show that, within admissible and dynamically active regimes, transition accessibility is not uniform but exhibits a structured, directional organization in perturbation space. Accessibility depends strongly on perturbation direction and cannot be reduced to scalar measures such as distance, energy, or probability. Using numerical experiments on canonical chaotic systems, we demonstrate that reachability is characterized by anisotropic directional dependence, heavy-tailed first-passage time distributions, and fractal accessibility boundaries. These results indicate that transition accessibility is governed by the geometric structure of system trajectories rather than by scalar properties of state space. We further demonstrate that this structure has empirical consequences. Using real-world climate data (ENSO), we construct a minimal structural signal capturing local destabilization and show that transitions are systematically preceded by detectable changes in system structure. The proposed framework achieves improved predictive precision relative to baseline and random models, with statistically significant performance. These findings establish transition accessibility as a measurable and operational property of dynamical systems. The results imply that admissibility does not guarantee accessibility, and that scalar descriptions of transition accessibility are fundamentally insufficient. Structured Reachability thus provides a bridge between dynamical structure and empirical predictability, enabling identification of transition-prone states prior to observed events.