Attractors of Singularly Perturbed Parabolic Systems of First Degree of Nonroughness in a Plane Domain

We study the problem of the attractors of the boundary-value problem
$$ u_t=\sqrt \varepsilon (D_0 + \sqrt \varepsilon D_1)\Delta u + (A_0 + \varepsilon A_1)u + F(u),\qquad u_x|_{x=0,x=l_1} = u_y|_{y=0,y=l_2}=0, $$
where $0\le\varepsilon\ll 1$, $u\in \mathbb{R}^N$, $N\ge 3$, $\Delta $ is the Laplace operator, and $-D_0$ is the Hurwitz matrix. For such a boundary-value problem, under certain assumptions, we establish the existence of any finite fixed number of stable cycles, provided that $\varepsilon>0$ is chosen appropriately small. In other words, it is shown that this boundary-value problem involves the buffer phenomenon.