A Unified Information‑Geometric Framework for Chaos and Catastrophe Theory

Nonlinear systems exhibit two fundamental forms of complexity: 1. sudden structural transitions (catastrophes) 2. sustained dynamical unpredictability (chaos) Although both arise from nonlinear mechanisms, they have traditionally been studied separately. This paper introduces a unified framework integrating catastrophe geometry, chaotic dynamics, and information theory. Key results: 1. Catastrophe points correspond to Fisher information singularities. 2. Chaotic regimes exhibit positive Lyapunov exponents and positive metric entropy. 3. Information theory provides the quantitative bridge linking geometric sensitivity and dynamical unpredictability. This yields a triadic structure: 1. Geometry — catastrophe set and Fisher metric 2. Dynamics — attractors, Lyapunov exponents 3. Information — entropy, complexity, sensitivity This unified perspective offers a comprehensive description of nonlinear systems and supports broader conceptual interpretations relevant to physics, mathematics, and metaphysics.