Pretzel Orbits in Simulations of Epitaxial Crystal Growth

Abstract

A third order autonomous ordinary differential equation is studied that describes stationary solutions of a nonlinear partial differential equation. The PDE models the growth of an epitaxial film on misoriented crystal substrates and is similar to the Kuramoto–Sivashinsky equation, but contains an additional nonlinear term. The equilibria, the periodic solutions, and the heteroclinic orbits of the ODE are analyzed, and stability results are given. Parameter regions are identified where the equilibria and the periodic solutions are unstable, but other bounded solutions exist. Their phase portrait is a double focus ("pretzel") that connects the stable and the unstable manifolds of the equilibria.