Delay Differential Equations

Delay differential equations (DDEs) are similar to ordinary differential equations, except that they involve past values of the dependent variables and/or their derivatives. Because of this, rather than needing an initial value to be fully specified, DDEs require input of an initial history (sequence of values) instead. Typically, these initial (history) functions are not fully compatible with the model dynamics, leading to discontinuities as the method switches from the initial function to values recorded through the integration. Whether the delay is introduced in the values or in the derivatives has great implications for the propagation in time of the discontinuities.