Consider the nonlinear system

$x^{\text{'}}=x+5y-x^3-x{y}^2-4y^3$

$y^{\text{'}}=-5x+y-x^{2}y+4x{y}^2-y^3$

Consider the polar coordinates $x=rcos()$ and $y=rsin().$

$($a$)$ Compute $r^{\text{'}}$ as a function of $(r,).$ In your answer use $t$ in place of $.$

$($b$)$ Compute ${}^{\text{'}}$ as a function of $(r,).$ In your answer use $t$ in place of $.$

$($c$)$ By using $($a$)$ and $($b$),$ conclude where the solution curves with $(x_0,y_0)(0,0)$ approach as $t.$

Enter your answer as a symbolic

function of $r,t,$ as in these

examples

Enter your answer as a symbolic

function of $r,t,$ as in these

examples

$($A$)$ diverge to infinity. $($B$)$ limit cycle at the unit circle traversed in the clockwise direction

$($C$)$ one of the two two stable equilibrium points on the unit circle. $($D$)$ the stable equilibrium point at $(0,0)$

$($E$)$ limit cycle at the unit circle traversed in the counter$-$clockwise direction

$($F$)$ the stable equilibrium point on the unit circle.