timecontinuous

Timecontinuous is a term used to describe processes, signals, or models that evolve in continuous time, as opposed to discrete-time evolution. In standard usage, the preferred terms are continuous-time or time-continuous, though some contexts may encounter the fused form timecontinuous.

In mathematics and engineering, a time-continuous signal is a function x(t) defined for all real-valued time t (or for t in a nonnegative interval). Continuous-time dynamical systems are commonly described by differential equations such as dx/dt = f(x,t); the system output may be given by y(t) = h(x,t) or y(t) = Cx + Du in state-space form. This framework contrasts with discrete-time models, where time advances in steps t = 0, 1, 2, …

In probability and stochastic processes, a continuous-time process X(t) has an index set that is a continuum, typically t ≥ 0. Canonical examples include Brownian motion and other diffusion processes, as well as Poisson and other counting processes that evolve in continuous time. Sample paths are real-valued or vector-valued functions of real time.

Applications of time-continuous models span physics, engineering, economics, biology, and control theory. They are analyzed using tools such as differential equations, Laplace and Fourier transforms, and continuous-time stochastic calculus. When implementing simulations, continuous-time models are often discretized via time-stepping methods (e.g., Euler, Runge-Kutta) or solved exactly when possible.

Key distinctions include the nature of time (continuous versus discrete), representation (differential equations versus difference equations), and the associated analytical and numerical techniques. Timecontinuous thus denotes the domain of time over which a system or signal is defined and studied.