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Perturb the sinusoidal cellular flow by $ϵ\cos x\cos y\sin (\omega t)$ in its stream function. On $y=0$, parameterizing the unperturbed orbit with $q_0(0)=(X_0,0)$ gives $x(s)=2\arctan e^{s+a}$, $a=\log \tan (X_0/2)$. The heteroclinic Melnikov function for a periodic planar flow has integrand ${\mathrm{sech}}^2(t-t_0+a)\sin (\omega t)$. Its odd part vanishes and the Fourier transform of ${\mathrm{sech}}^2$ gives the displayed result. For each fixed nonzero frequency its simple zeros produce transverse intersections at sufficiently small perturbation, hence exchange of fluid lobes across the former cell boundary. The leading splitting is exponentially small at high frequency; that estimate is not uniform in frequency and perturbation size.

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.