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In mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the functional state, i.e. partial differential equations which are infinite dimensional, as opposed to ordinary differential equations having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:
A general form of the time-delay differential equation for x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} is