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September 28th, 2010, 03:43 PM   #1
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Dynamical Systems Question.

Hello, I have a tough dynamical systems question and I was hoping someone could point me in the right direction.


Let be the flow of the differential equation for and let be the flow of the differential equation .

Verify that for and. What can you say about the comparisons of solutions for . Also, show that the trajectories of go to infinty in finite time.


i really do not know where to begin. Can someone help me out at all? I tried representing the trajectories with the fundamental theorem of calculus, but I dont know where to go, or even if thats the right idea. Thanks

Richard.
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September 28th, 2010, 07:34 PM   #2
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Re: Dynamical Systems Question.

Note that for any , which means that 1) both and are strictly monotonically increasing for all t, and that 2) for all t > 0 we have because the derivative of is strictly larger than that of .

We can actually solve the second differential equation explicitly using the standard technique of separation of variables:

But the integral on the left hand side gives , adding to both sides, and then taking the tangent of both sides gives us that:
.

The tangent function becomes infinite at so becomes infinite at which is finite. Since the derivative for the other flow is strictly larger than the one we explicitly solved for (that is for ) we know that it must also go to infinity in finite time as it grows faster than a function we already showed exhibits that behavior.

I'm not sure about what they're looking for as far as the different initial conditions go, but here is some food for thought:
The value of , and is an odd function, so that and

Hope this helps a bit
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September 28th, 2010, 07:56 PM   #3
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Re: Dynamical Systems Question.

It does help a bit, but I am still confused as to how you are certain that what you listed as 1) and 2) hold true. What makes them true, exactly? Also, how do I show its true for ? This is getting more confusing by the minute
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