Mathematically, the problem considered here is that of heteroclinic connections for a system of two second-order differential equations of gradient type, in which a small parameter $\epsilon$ conveys a singular perturbation. The physical motivation comes from a multi-order-parameter phase field model, developed by Braun et al.  and , for the description of crystalline interphase boundaries. In this, the smallness of $\epsilon$ is related to large anisotropy. The mathematical interest stems from the fact that the smoothness and normal hyperbolicity of the critical manifold fails at certain points. Thus the well-developed geometric singular perturbation theory ,  does not apply. The existence of such a heteroclinic, and its dependence on $\epsilon$, is proved via a functional analytic approach.
P. C. Fife. C. Sourdis. "Existence of heteroclinic orbits for a corner layer problem in anisotropic interfaces." Adv. Differential Equations 12 (6) 623 - 668, 2007. https://doi.org/10.57262/ade/1355867448