Use the definition of stability of equilibrium to prove (1,0) is unstable.Given the systemx′=x(1−r)+y(x−r)y′=y(1−r)+x(r−x)where r2=x2+y2. I don't know how to use the definition of stability of equilibrium to show that the equilibrium (1,0) is unstable.

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Use the definition of stability of equilibrium to prove (1,0) is unstable.Given the systemx′=x(1−r)+y(x−r)y′=y(1−r)+x(r−x)where r2=x2+y2. I don&#039;t know how to use the definition of stability of equilibrium to show that the equilibrium (1,0) is unstable.

Sabrina Campos · Accepted Answer

Explanation:You can change the ODE from the cartesian coordinate to the polar cordinate:rr′=×′+yy′. Therefore, rr′=(x2+y2)(1−r)=r2(1−r), and so r′=r(1−r).We can conclude that:For 0&amp;lt;r&amp;lt;1,r′&amp;gt;0 and r is increasing.For r&amp;gt;1,r′&amp;lt;0 and r is decreasing.Hence, we approach the unit circle in this system, and it is wrong to say the equilibrium (1,0) is unstable.

Use the definition of stability of equilibrium to

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