ratedepend

Ratedepend is a concept used in mathematical and computational modeling to describe systems whose observable response depends on the rate at which an input changes, rather than only on the input’s instantaneous value. In ratedepend formulations, the output y is influenced by both the state variable x and its time derivative dx/dt, allowing the model to capture rate-sensitive effects such as inertia, damping, accommodation, or saturation that unfold over time.

A typical simple form is y(t) = a + b x(t) + c dx/dt, with a, b, c as parameters. More general versions include nonlinear terms, memory kernels, or multiple derivative terms. In practice, ratedepend models are distinguished from rate-independent models by their ability to produce different responses under the same x with different rates of change.

Applications span engineering (friction vs velocity, actuation lag), pharmacology (drug effect linked to administration rate), ecology (resource fluctuation responses), economics (reaction to rapid price changes), and materials science (rate-dependent deformation).

Estimation and validation require careful handling of data; derivatives are sensitive to noise, so smoothing or specialized techniques are often used. Model selection balances complexity and interpretability, and identifiability can be challenging with limited data.

See also: rate dependence, rate-dependent deformation, hysteresis, dynamic systems.