Solve Time Delay Equations: Derive (6) and (7) from Mackey-Glass

[bor0000]

i need to solve an equation of the form (5) from this webpage http://chaos.phy.ohiou.edu/~thomas/chaos/mackey-glass.html
first it asks to "derive the corresponding eigenvalue equation"- i presume they mean to derive (6)? if so, i don't know how they get it.
and then they ask for something similar to deriving (7), i have no idea how to proceed... thanks.

 

[saltydog]

i need to solve an equation of the form (5) from this webpage http://chaos.phy.ohiou.edu/~thomas/chaos/mackey-glass.html
first it asks to "derive the corresponding eigenvalue equation"- i presume they mean to derive (6)? if so, i don't know how they get it.
and then they ask for something similar to deriving (7), i have no idea how to proceed... thanks.

Jesus dude, I gotta' get that book! One way to proceed of course is . . . to check out the book from a library. Deriving the eigenvalue equation is just substituting the assumed solution:

$$y(t)=e^{\lambda t}$$

into the "linearized equation:

$$y^{'}=\alpha y+\beta y_{\tau}$$

remembering [itex]y_{\tau}=e^{\lambda(t-\tau)}[/tex]<br /> <br /> Deriving (7) I assume means to solve for lambda in:<br /> <br /> $$\lambda=\alpha+\beta e^{-\lambda \tau}$$<br /> <br /> assuming lambda is complex and determining under what conditions the real part is less than zero (not sure though, just my assumption). Really, this would take me days to fully study, a week maybe. But very interesting. Thanks. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" />[/itex]

 

[saltydog]

You know we can make progress with this. First put it into standard DDE form:

$$\frac{dW}{dt}=aW(t-\tau)\frac{k_1}{k_1+[W(t-\tau)]^n}-bW$$

That gives the rate of change of the density of white blood cells circulating in the blood as a function of the current density as well as the density at a previous time. This is doable. First thing to note that with DDEs, rather than an initial point given as the initial condition, an initial function has to be given in the interval:

$$(-\tau,0)$$

So we'll call that initial condition $f_1(t)$

Now, in the interval $(0,\tau)$, we have a regular ODE:

$$\frac{dW}{dt}=af_1(t-\tau)\frac{k_1}{k_1+[f_1(t-\tau)]^n}-bW$$

Which we can integrate from 0 to $\tau$. We'll call that function $f_2(t)$. Now, plug that into the DDE:

$$\frac{dW}{dt}=af_2(t-\tau)\frac{k_1}{k_1+[f_2(t-\tau)]^n}-bW$$

and integrate from $\tau$ to $2\tau$. See what's happening? Keep doing that. It get's messy. And how can this DDE help model the onset of lukemia? Think I'll spend some time on it.

 

[saltydog]

Here's an example.

$$\frac{dy}{dt}=\frac{0.2 y(t-14)}{1+y(t-14)^{10}}-0.1y(t);\quad y(<0)=0.5$$

As I stated earlier, we'll integrate in intervals of the delay using the previous solution as the delay functions in the ODE. Here is the first interval in Mathematica:

$$f_0[t]=0.5;$$

$$\text{sol1=NDSolve}[{y^{'}==\frac{0.2 f_0[t-14]}{1+f_0[t-14]^{10}}-0.1y(t),y[0]=f_0[0]},y,\{t,0,14\}]$$

$$f_1[t\_]\text{:=Evaluate[y[t]/.Flatten[sol1]];}$$


Here's the second one:

$$\text{sol2=NDSolve}[{y^{'}==\frac{0.2 f_1[t-14]}{1+f_1[t-14]^{10}}-0.1y(t),y[14]=f_1[14]},y,\{t,14,28\}]$$

$$f_2[t\_]\text{:=Evaluate[y[t]/.Flatten[sol2]];}$$

Note how I substituted $f_1(t-14)$ into the ODE to represent the delay for the second interval. Here's the third interval:

$$\text{sol3=NDSolve}[{y^{'}==\frac{0.2 f_2[t-14]}{1+f_2[t-14]^{10}}-0.1y(t),y[28]=f_2[28]},y,\{t,28,42\}]$$

$$f_3[t\_]\text{:=Evaluate[y[t]/.Flatten[sol3]];}$$

And so on for each interval. A plot of the first three intervals is attached. Is there another way to do this? Should I go over to the ODE forum and ask as this isn't my homework?

 

[bor0000]

thank you!