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 September 29th, 2017, 03:03 PM #1 Newbie   Joined: Sep 2017 From: Moscow Posts: 4 Thanks: 0 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.
 September 30th, 2017, 12:47 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,089 Thanks: 1902 What was the exact wording of the original question?
 October 1st, 2017, 02:34 AM #3 Newbie   Joined: Sep 2017 From: Moscow Posts: 4 Thanks: 0 Explore the stability in the neighbourhood of zero
 October 1st, 2017, 02:44 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,089 Thanks: 1902 The stability of what?
October 1st, 2017, 08:53 PM   #5
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Quote:
 Originally Posted by oscartempter 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.

October 2nd, 2017, 02:28 AM   #6
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Quote:
 Originally Posted by skipjack 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

October 2nd, 2017, 02:30 AM   #7
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Quote:
 Originally Posted by SDK $r$ 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.

 October 6th, 2017, 09:35 AM #8 Senior Member   Joined: Sep 2016 From: USA Posts: 531 Thanks: 304 Math Focus: Dynamical systems, analytic function theory, numerics 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
 October 27th, 2017, 01:02 PM #9 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 Why in the world would you go from a specified function to a differential equation and then ask about the stability of that solution to the differential equation? $\displaystyle r= \sqrt{x^2+ y^2}$ is the same as $\displaystyle x^2+ y^2= r^2$ so the trajectories are circles around the origin. The origin is a stable (but not asymptotically stable) equilibrium. Thanks from SDK

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