Dynamical System (timtam's question at Yahoo Answers)

[Fernando Revilla]

Here is the question:

For the system:
xdot= -y -xsqrt(x^2+y^2)
ydot= x -ysqrt(x^2+y^2)
Find a trapping neighbourhood for the origin.

Here is a link to the question:

Trapping neighbourhood (dynamical systems)? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello timtam,

On $x^2+y^2=1$ we have $$(x,y)\cdot v(x,y)=(x,y)\cdot \left(-y -x\sqrt{x^2+y^2},x -y\sqrt{x^2+y^2}\right)=\ldots=-2<0$$ That is, on the boundary of the closed set $R\equiv x^2+y^2\le 1$ the vector field is pointing towards the interior of $R$ so, the open unit disk is a trapping region.