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

@@ -6,3 +6,12 @@ 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 $t=0$
 
+## Method of Steps
+Think of DDE as being a mapping between the past interval and the present interval
+$$x(t) = \phi_{i-1}(t)$$ on any interval $[t_i-1,t_i]$
+$$\int_{\phi_{i-1}(t)}^{x(t)}dx' = - \int_{t_i}^t \phi_{i-1}(t' - 1)dt'$$
+Then after some steps
+$$\frac{dx}{dt} = -x(t-1) \rightarrow dx' = -x'(t'-1)dt'$$
+And then this can be solved at each interval.
+- This gets annoying.
+- Need to solve at each interval over and over.