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(i) Show that the equations
$\frac{d u_{1}}{d t}=u_{1}-u_{2}-u_{1}\left(u_{1}^{2}+u_{2}^{2}\right), \quad \frac{d u_{2}}{d t}=u_{1}+u_{2}-u_{2}\left(u_{1}^{2}+u_{2}^{2}\right)$
have their only fixed point at $(0,0)$.
(ii) Show that the origin is an unstable focus for the linear approximation and also for the nonlinear system. (iii) Express (1) in polar coordinates and solve the system of differential equations.
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