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September 29th, 2017, 02:03 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, 11:47 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,845 Thanks: 1567 
What was the exact wording of the original question?

October 1st, 2017, 01:34 AM  #3 
Newbie Joined: Sep 2017 From: Moscow Posts: 4 Thanks: 0 
Explore the stability in the neighbourhood of zero

October 1st, 2017, 01:44 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,845 Thanks: 1567 
The stability of what?

October 1st, 2017, 07:53 PM  #5  
Senior Member Joined: Sep 2016 From: USA Posts: 357 Thanks: 194 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
 
October 2nd, 2017, 01:28 AM  #6 
Newbie Joined: Sep 2017 From: Moscow Posts: 4 Thanks: 0  
October 2nd, 2017, 01:30 AM  #7  
Newbie Joined: Sep 2017 From: Moscow Posts: 4 Thanks: 0  Quote:
To get this equation was one of the problems, then I need to explore it.  
October 6th, 2017, 08:35 AM  #8 
Senior Member Joined: Sep 2016 From: USA Posts: 357 Thanks: 194 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, 12:02 PM  #9 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,102 Thanks: 850 
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.


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