A Morse--Smale height function on a tilted torus

The negative-gradient equations are $\dot{u}=-\cos u\mathrm{/}(SA)$ and $\dot{v}=(\sin v\sin u-\epsilon \cos v)\mathrm{/}S$. The field is complete on this compact torus. Critical points require $u=\pi \mathrm{/}2$ or $3\pi \mathrm{/}2$. For $u=\pi \mathrm{/}2$ they have $v=\varphi ,\pi +\varphi$; for $u=3\pi \mathrm{/}2$ they have $v=2\pi -\varphi ,\pi -\varphi$. The mixed Hessian entry is zero, the $uu$ entry is $-A\sin u\mathrm{/}S$, and the $vv$ entry is $-(\cos v\sin u+\epsilon \sin v)\mathrm{/}S$, which equals $\pm 1$ at these points. Thus all four are nondegenerate: a maximum, an upper saddle $a=(\pi \mathrm{/}2,\pi +\varphi )$, a minimum, and a lower saddle $b=(3\pi \mathrm{/}2,\pi -\varphi )$, respectively. Their saddle values are $(2-S)\mathrm{/}S$ and $-(2-S)\mathrm{/}S$.

The closed strip $\pi \le v\le 2\pi$ is forward invariant: on its lower boundary $\dot{v}=\epsilon \mathrm{/}S>0$, and on its upper boundary $\dot{v}=-\epsilon \mathrm{/}S<0$. Since $a$ lies in its interior, every orbit with backward limit $a$ stays in this strip, and cannot have forward limit $b$, which is outside it. The reverse connection is excluded by strict decrease of $f$. Thus there are no connections between the distinct saddles.

On a surface all other nonempty intersections are automatically transverse: an unstable manifold of a maximum or stable manifold of a minimum is open; the remaining extremal stable/unstable manifolds are singletons and meet only their own complementary open manifold. At each saddle its stable and unstable tangent lines at that saddle are the complementary negative-gradient eigenspaces. A nonconstant orbit cannot have identical endpoints because $f$ strictly decreases. Together with step 2.1 these observations cover every pair and prove $(f,g)$ Morse--Smale.

A connecting trajectory has $df(X)<0$, so its regular-level slice is a transverse hypersurface in the parametrized intersection. By [F2] and [F3] the space modulo time has dimension $\lambda (p)-\lambda (q)-1$. This is zero for index drop one and one for the maximum-to-minimum pair. In particular, the latter slice is not a finite set: the open basins of the maximum and minimum overlap, since their complements are the finitely many saddle separatrices and critical points.