Consider the two dimensional map [x_{n+1}=alphaleft(x_{n}-frac{1}{4}left(x_{n}+y_{n} ight)^{2} ight) quad, quad y_{n+1}=frac{1}{alpha}left(y_{n}+frac{1}{4}left(x_{n}+y_{n} ight)^{2} ight)] that approximates the chaotic

Consider the two dimensional map

\[x_{n+1}=\alpha\left(x_{n}-\frac{1}{4}\left(x_{n}+y_{n}\right)^{2}\right) \quad, \quad y_{n+1}=\frac{1}{\alpha}\left(y_{n}+\frac{1}{4}\left(x_{n}+y_{n}\right)^{2}\right)\]

that approximates the chaotic scattering behavior of a projectile off a region near the origin where it collides with a bunch of targets. Fix \(\alpha=5\) and \(y_{0}\) to some small value near the origin. Then start with a bunch of values for \(x_{0}\) near the origin but positive, and compute the number of steps \(S\left(x_{0}\right)\) it takes for the projectile to leave the collision basin, say when \(x_{n}<-5\). Plot \(S\left(x_{0}\right)\).