926 B
926 B
\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
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.
Stability of DDEs
For ODE: equilibrium point is 0 of derivative For DDE: dx/dt is still 0