timevarying

Timevarying refers to phenomena whose characteristics or parameters change over time. In mathematics and engineering, a timevarying function or system depends explicitly on time t, rather than being defined solely by the current state or input. By contrast, time-invariant models maintain fixed parameters that do not depend on t.

In signal processing, a timevarying (or time-variant) signal changes its form over time, such as a frequency-modulated wave or an amplitude that evolves. In control theory, timevarying systems have dynamics that depend on time, often expressed in state-space form as x'(t) = A(t) x(t) + B(t) u(t), y(t) = C(t) x(t) + D(t) u(t). If the coefficient matrices A, B, C, D are constant, the system is time-invariant; if they depend on t, the system is timevarying.

Timevarying parameters also appear in modeling, where coefficients, stiffness, resistance, or other properties change with time due to aging, temperature, or adaptation. Examples include a resistor with R(t) changing over time, or a mechanical system with time-dependent stiffness k(t), or an adaptive controller that updates gains dynamically.

Analysis methods for timevarying systems include differential equations with explicit time dependence, piecewise-linear or piecewise-constant approximations, and specialized theories such as Floquet analysis for periodic time variation.

In practice, recognizing timevariability is crucial for accurate modeling, estimation, and control. Timevarying models capture nonstationary behavior across engineering, physics, economics, and related fields. See also time-invariant, non-stationary processes, adaptive control.