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September 29th, 2017, 02:03 PM   #1
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Question Explore the stability in the neighbourhood of zero

Help to explore an equation in spherical coordinates. First of all, it was the following equation $\displaystyle r(t)=\sqrt{x^{2}(t)+y^2(t)}$
then I differentiated it $\displaystyle \dot r(t)=\frac{x(t)\dot x(t)+y(t)\dot y(t)}{\sqrt{x^{2}(t)+y^{2}(t)}} $ and I don't know what to do next. I know that I should use Lyapunov functions in order to explore the stability but I can do it only for a system of equations in Cartesian coordinates... Trying to transit to Cartesian coordinates sufficiently complicates derivatives... So I believe there is must be a simple way to do it in spherical coordinates.
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September 30th, 2017, 11:47 AM   #2
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What was the exact wording of the original question?
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October 1st, 2017, 01:34 AM   #3
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Explore the stability in the neighbourhood of zero
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October 1st, 2017, 01:44 AM   #4
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The stability of what?
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October 1st, 2017, 07:53 PM   #5
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Quote:
Originally Posted by oscartempter View Post
Help to explore an equation in spherical coordinates. First of all, it was the following equation $\displaystyle r(t)=\sqrt{x^{2}(t)+y^2(t)}$
then I differentiated it $\displaystyle \dot r(t)=\frac{x(t)\dot x(t)+y(t)\dot y(t)}{\sqrt{x^{2}(t)+y^{2}(t)}} $ and I don't know what to do next. I know that I should use Lyapunov functions in order to explore the stability but I can do it only for a system of equations in Cartesian coordinates... Trying to transit to Cartesian coordinates sufficiently complicates derivatives... So I believe there is must be a simple way to do it in spherical coordinates.
$r$ is a transformation but you have not specified the differential equation you are trying to study. The fact that you seem to not be aware of this indicates you don't have a great grasp on what the question is even asking. Try posting a screenshot of the original question in your textbook or wherever it comes from.
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October 2nd, 2017, 01:28 AM   #6
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Originally Posted by skipjack View Post
The stability of what?
the stability of differential equation that I got after differentiating:
$\displaystyle \dot r(t)=\frac{x(t)\dot x(t)+y(t)\dot y(t)}{\sqrt{x^{2}(t)+y^{2}(t)}} $ in the neighbourhood of zero)) I seem it means in the area of t=0
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October 2nd, 2017, 01:30 AM   #7
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Quote:
Originally Posted by SDK View Post
$r$ is a transformation but you have not specified the differential equation you are trying to study. The fact that you seem to not be aware of this indicates you don't have a great grasp on what the question is even asking. Try posting a screenshot of the original question in your textbook or wherever it comes from.
Don't you see? I specified $\displaystyle \dot r(t)=\frac{x(t)\dot x(t)+y(t)\dot y(t)}{\sqrt{x^{2}(t)+y^{2}(t)}}$

To get this equation was one of the problems, then I need to explore it.
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October 6th, 2017, 08:35 AM   #8
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Just to reiterate, your question makes no sense.

The discussion of stability applies to invariant sets in phase space. Therefore, it makes no sense to talk about stability at $t = 0$ or stability at any "time" for that matter.

I suspect this question is asking you to explore the stability of an equilibrium solution of an ODE and that as a hint it suggests switching to spherical coordinates. Do you see that there is no way to determine the equilibria of your differential equation because we don't know $\dot{x}, \dot{y}$ since these should be specified by some vector field which you aren't telling us.

I suggest reading the following:
http://www.math.psu.edu/tseng/class/...0ODE%20pt2.pdf
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