Suppose, we have a system described by a set of ordinary differential equations. Is it possible to approximate the system with delay differential equations such that the properties of the system is preserved.
CASE 1: dx1/dt = -a*x1 + b*x2
dx2/dt = -k*x2 + k*x1, where a, k=constant, x1, x2: states; x2 is say a 10 min(k=0.1 min^-1) delayed version of x1. Can we approximate the above equation by dx1/dt = -a*x1 + b*x2(t-T), where T:delay parameter
CASE 2: dx1/dt = -a*x1 + b*x2
dx2/dt = -c*x2 + d*x3
dx3/dt = -e*x3 + u, where x1,x2,x3: states, u: input, a,b,c,d,e: constants. Can we approximate these system by delay differential equations.
@Ryan Vogt
Thanks for your reference. But can you suggest any examples of that in some reference.
Typically in the literature, folks tend to go from DDE to ODE. This is done by a change of variables. You can think that you can go from ODE to DDE by new change of variables. For example, you can say t=t1-t2. Call t1 your new notion of time, and t2 to be the delay variables. Where t1 >= t2. You can now pose your differential equations as functions of the independent variables t,t1, and t2.
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