Lorenz equations part II: "randomly" rotated homoclinic orbits and chaotic trajectories

$x' =s(y-x), \quad y'= Rx -y-xz, \quad z'= xy -qz,$

where $s$, $R$, and $q$ are positive parameters. We show by a purely analytic proof that for each non-negative integer $N$, there are positive parameters $s, q, $ and $R$ such that the Lorenz system has homoclinic orbits associated with the origin (i.e., orbits that tend to the origin as $t\to \pm \infty$) which can rotate around the $z$-axis $N/2$ times; namely, the $x$-component changes sign exactly $N$ times, the $y$-component changes sign exactly $N+1$ times, and the zeros of $x$ and $y$ are simple and interlace.