I Want YOU To Solve This

For any value of $y$ , this problem defines a dynamic system $f(u) = \frac{u^2 + y^2)} 2.$ Fixed points $f(u) = u$ are found when $u^2 + y^2 = 2u\ \ \ \therefore\ \ \ u = 1 \pm \sqrt{1 - y^2}.$ There are no fixed points for $y \geq 1$ , so that for these values there is no convergence.

A fixed point is an attractor if $|df/du| < 1$ ; since the derivative is $\frac{df}{du} = u,$ the fixed point $u_1 = 1 - \sqrt{1 - y^2}$ attracts but $u_2 = 1 + \sqrt{1 + y^2}$ repels. Consider now the basins of attraction:

• Positive $u$ are attracted to $u_1$ if $u < u_2$ , but are repelled to $+\infty$ if $u > u_2$ .

• Negative $u$ behave precisely like their absolute values $|u|$ .

We therefore expect convergence precisely if $|y| \leq 1$ and $|x| \leq 1 + \sqrt{1 + y^2}$ . This region may be viewed as the union of the square $[-1,1] \times [-1,1]$ and two semi-circles of radius 1, defined by $|x| > 1$ and $(x\pm 1)^2 + y^2 \leq 1^2$ . The area of the square is 4, and that of the two semicircles is $\pi$ , so that the answer is $\boxed{4 + \pi}$ .