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

In summary, the author is asking for help solving an equation of the form (5) from this webpage http://chaos.phy.ohiou.edu/~thomas/chaos/mackey-glass.html . The first thing to note is that in order to solve for the eigenvalue equation, the author needs to derive the corresponding eigenvalue equation. The author also notes that this can be done, but it gets messy. The author suggests that instead, he study the equation for an extended period of time.
  • #1
bor0000
50
0
i need to solve an equation of the form (5) from this webpage http://chaos.phy.ohiou.edu/~thomas/chaos/mackey-glass.html [Broken]
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.
 
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  • #2
bor0000 said:
i need to solve an equation of the form (5) from this webpage http://chaos.phy.ohiou.edu/~thomas/chaos/mackey-glass.html [Broken]
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:

[tex]y(t)=e^{\lambda t}[/tex]

into the "linearized equation:

[tex]y^{'}=\alpha y+\beta y_{\tau}[/tex]

remembering [itex]y_{\tau}=e^{\lambda(t-\tau)}[/tex]

Deriving (7) I assume means to solve for lambda in:

[tex]\lambda=\alpha+\beta e^{-\lambda \tau}[/tex]

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. :smile:
 
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  • #3
You know we can make progress with this. First put it into standard DDE form:

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

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:

[tex](-\tau,0)[/tex]

So we'll call that initial condition [itex]f_1(t)[/itex]

Now, in the interval [itex](0,\tau)[/itex], we have a regular ODE:

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

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

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

and integrate from [itex]\tau[/itex] to [itex]2\tau[/itex]. 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.
 
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  • #4
Here's an example.

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

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:

[tex]f_0[t]=0.5;[/tex]

[tex]\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\}][/tex]

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


Here's the second one:

[tex]\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\}][/tex]

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

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

[tex]\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\}][/tex]

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

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?
 

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  • #5
thank you!
 

What is a time delay equation?

A time delay equation is a mathematical model that describes the behavior of a system over time, taking into account the effects of a time delay in the system's response to changes in input.

What is the Mackey-Glass model?

The Mackey-Glass model is a type of time delay equation that was proposed by mathematician Michael Mackey and physiologist Leon Glass in the 1970s. It is commonly used to study phenomena in biology, chemistry, and economics.

How do you derive equation (6) from the Mackey-Glass model?

Equation (6) is derived from the Mackey-Glass model by using differential calculus and the chain rule to express the time derivative of the system's output in terms of the system's input and state variables.

What is the significance of equation (7) in the Mackey-Glass model?

Equation (7) is significant because it represents the delayed feedback in the system, where the current output is influenced by past outputs. This delayed feedback is a key feature of the Mackey-Glass model and is used to study the effects of time delays on system behavior.

What are some applications of solving time delay equations in real-world systems?

Solving time delay equations, such as the Mackey-Glass model, can help us understand and predict the behavior of complex systems with delayed responses, such as population dynamics, chemical reactions, and economic systems. This can have practical applications in fields such as medicine, engineering, and finance.

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