A Generalized Template Equation for Energy Dynamics Across Complex Systems

This preprint introduces a generalized template differential equation designed to model energy dynamics across diverse systems—including physical, biological, social, and artistic domains. The central equation takes the form:

a(X) X'' - b(X) X' + c(X) X^n = S(t, X)

where X(t) represents an abstract energy-like state variable, and S(t, X) denotes a flexible driving function analogous to a force term in Newtonian dynamics. This structure enables unification of various phenomena under a single mathematical framework, allowing for AI-assisted synthesis of governing dynamics through the automated generation of S 

We classify the system's behavior into different regimes—stable, oscillatory, divergent, or chaotic—based on the properties of S , the nonlinear power n , and the coefficient functions. This framework is intended to facilitate both theoretical analysis and applied modeling of real-world phenomena such as emotional responses, artistic stimulation, policy evolution, and systemic collapse.

The manuscript also outlines the potential for a generative AI system (S-forming engine) to construct problem-specific ssource terms, enablig semi-automated system modeling.