Leonid Shilnikov and mathematical theory of dynamical chaos

The paper states that, near a transverse Poincaré homoclinic orbit to a saddle periodic orbit, the set of orbits entirely contained in a small neighborhood is hyperbolic and is in one-to-one correspondence with bi-infinite sequences of two symbols; this is presented as Shilnikov’s 1967 result and contrasted with Smale’s 1965 horseshoe under linearization assumptions . The model’s solution constructs the same object via a standard Smale–Birkhoff horseshoe for a Poincaré return map and then suspends to the flow, yielding a conjugacy to the suspension over the full two-shift. This matches the paper’s claim. The approaches differ in methods (Shilnikov’s boundary-value approach versus the model’s horseshoe/Markov rectangles plus suspension), but the conclusions coincide. Minor caveats (e.g., coding on rectangle boundaries, dimension of the section) are standard and easily fixed.