 Solution

The unperturbed sinusoidal cellular flow hasIts 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 $\mathrm{diag}(s,-s)$ 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 $\left(\begin{array}{cc}0& -s\\ s& 0\end{array}\right)$, where now $s=\sin x\sin y=\pm 1$. 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.