20 lines
926 B
Markdown
20 lines
926 B
Markdown
$$ \frac{dx}{dt} = \vec f(x(t), x(t-\tau), x(t-\tau_2), ..., x(t-\tau_n)) $$
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Example:
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First order DE
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$$ \frac{dx}{ct} = -x \rightarrow x(t) = x_0e^{-t}$$
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First order DDE:
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$$\frac{dx}{dt} = -x(t-1)$$
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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 $t=0$
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## Method of Steps
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Think of DDE as being a mapping between the past interval and the present interval
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$$x(t) = \phi_{i-1}(t)$$ on any interval $[t_i-1,t_i]$
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$$\int_{\phi_{i-1}(t)}^{x(t)}dx' = - \int_{t_i}^t \phi_{i-1}(t' - 1)dt'$$
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Then after some steps
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$$\frac{dx}{dt} = -x(t-1) \rightarrow dx' = -x'(t'-1)dt'$$
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And then this can be solved at each interval.
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- This gets annoying.
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- Need to solve at each interval over and over.
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## Stability of DDEs
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For ODE: equilibrium point is 0 of derivative
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For DDE: dx/dt is still 0 |