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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-12-02 Delay Differential Equation.md

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+$$ \frac{dx}{dt} = \vec f(x(t), x(t-\tau), x(t-\tau_2), ..., x(t-\tau_n)) $$
+Example:
+First order DE
+$$ \frac{dx}{ct} = -x \rightarrow x(t) = x_0e^{-t}$$
+First order DDE:
+$$\frac{dx}{dt} = -x(t-1)$$
+This is a problem. We cannot use an initial value, we need **an initial history function (IHF)**. This is the behaviour of x(t) defined in an interval $[-\tau_0, 0]$, assuming solution time starts at $t=0$
+