MHB Dynamical System (timtam's question at Yahoo Answers)

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The discussion focuses on finding a trapping neighborhood for the origin in a given dynamical system defined by the equations xdot = -y - x√(x²+y²) and ydot = x - y√(x²+y²). It is established that on the boundary defined by x²+y²=1, the vector field points inward, indicating stability. Specifically, the calculation shows that the dot product of the position vector and the velocity vector is negative, confirming that the open unit disk serves as a trapping region. This conclusion provides a clear understanding of the system's behavior near the origin. The findings are relevant for analyzing the stability of dynamical systems.
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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.
 
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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.
 
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