= Solution
The unperturbed <sinusoidal cellular flow> has
$$
\mathbf u_0=(\sin x\cos y,-\cos x\sin y),\qquad
\frac{d\psi_0}{dt}=\nabla\psi_0\cdot\mathbf u_0=0.
$$
Its <stream function> is a <first integral>, so each trajectory lies on a level set of $\psi_0$. This planar autonomous <Hamiltonian flow> has one degree of freedom and its conserved quantity reduces motion on regular levels to a quadrature: this is its integrability. The central cell has closed <streamlines>; the zero level is the grid of <separatrices>.
Solving both components of $\mathbf u_0=0$ gives <saddle equilibria> at $(n\pi,m\pi)$ and <center equilibria> at $(\pi/2+n\pi,\pi/2+m\pi)$. In the specified open square, the saddles are $(0,0),(\pi,0),(0,\pi),(\pi,\pi)$ and the only interior center is $(\pi/2,\pi/2)$. Centers at the outer square's corners are on its excluded boundary.
At a saddle the <Jacobian matrix> is $\operatorname{diag}(s,-s)$ with $s=(-1)^{n+m}$. If $n+m$ is even, the horizontal branches are unstable and vertical branches stable; if it is odd, those roles reverse. The branches extend along $y=m\pi$ and $x=n\pi$ and connect neighboring saddles as <heteroclinic orbits>. For example the bottom boundary flows from $(0,0)$ to $(\pi,0)$, the right boundary from $(\pi,0)$ to $(\pi,\pi)$, the top boundary from $(\pi,\pi)$ to $(0,\pi)$, and the left boundary returns to $(0,0)$. A connecting branch is unstable relative to its departing saddle and stable relative to its arriving saddle.
At a center the <Jacobian matrix> is $\begin{pmatrix}0&-s\\s&0\end{pmatrix}$, where now $s=\sin x\sin y=\pm1$. Its <eigenvalues> are $\pm i$. The local definite extremum of $\psi_0$ surrounds it by closed <streamlines>, giving a neutrally stable center, not an attracting equilibrium. Central circulation is counterclockwise and adjoining cells have alternating circulation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-46-cellular-flow.png]
{title=Unperturbed cellular streamlines, four interior saddles, and stable and unstable branches}
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