About the true nature of the graph: There are two totally different regimes for $t\ll-1$ and $t\gg1$, and there is a zone of transition in between, where nothing spectacular happens. For $t\ll-1$ one has $y(t)=e^t\doteq0$; therefore $r(t)\doteq(t,0,\cos t)\qquad (t\ll-1)\ ,$ which is an ordinary cosine curve in the left half of the $(x,z)$-plane, extending to $-\infty$ in the $x$-direction.
For $t\gg1$ it is obvious that $y\gg1$ is the prominent variable; therefore we choose $y$ as new parameter. In this way we obtain the new parametrization $\tilde r(y)=\bigl(\log y, y, \cos(\log y)\bigr)\qquad(y\gg1)\ .$ Looking from the point $(\infty,0,0)$ we see in the $(y,z)$-plane the curve $z=\cos(\log y)$ which is an oscillation in $z$-direction becoming ever slower as $y\to\infty$. At the same time the moving point on the space curve $y\mapsto\tilde r(y)$ is also increasing its $x$-coordinate towards $\infty$, but at ever decreasing speed.