Differentialequreging

Differentialequreging is a theoretical framework that combines differential equation models with an integrated regulatory dynamics component. It models systems in which a primary state evolves according to a differential equation while a separate regulatory module modulates inputs, parameters, or structure in response to the state. The regulatory module is itself described by a differential or hybrid equation, allowing continuous adjustment and, if needed, regime switches to reflect different operating modes.

Formally, a differentialequreging system can be described by a pair of coupled equations. The state x(t) evolves as dx/dt = f(x, r, t), where r(t) represents regulatory state variables or control signals. The regulation is governed by dr/dt = h(x, r, t) (or a piecewise/hybrid form with switching), and the actual input or parameters used by the state equation are functions of r, such as u = g(r, t) or parameters θ = θ(r, t). This coupling enables the model to capture feedback where regulation adapts in response to deviations from targets or changing conditions, including abrupt regime changes.

Applications of differentialequreging span biology, for example gene regulatory networks embedded in signaling models; engineering systems with adaptive or policy-driven regulation; economics and environmental models that incorporate regulatory responses. Analytical concerns include existence and uniqueness of solutions for the coupled system, stability under regulation, and identifiability of regulatory states from data. Computational approaches often employ time-stepping, hybrid system techniques, and parameter estimation tailored to the dual dynamics.

In relation to related concepts, differentialequreging sits at the intersection of differential equations, control theory, and hybrid dynamical systems, offering a framework for modeling regulated, adaptive dynamics in complex systems. See also: differential equation, control theory, hybrid systems, gene regulatory networks, adaptive control.