This monograph offers a compelling exploration of structures induced by smooth vector flows on the boundaries of compact manifolds, with a particular focus on flows that admit Lyapunov functions — strictly monotonic along each trajectory. These traversing flows reveal rich and intricate interactions between dynamics and geometry, especially at the manifold boundaries.At the heart of this work lies a central theme: how trajectories interact with boundaries, and how these interactions shape and are shaped by the manifold's topology. This perspective opens a natural gateway to a wide array of mathematical disciplines, including Singularity Theory, Combinatorics, Differential and Contact Geometry, Differential Topology, and Dynamical Systems.Written with a view toward both depth and interconnection, the monograph highlights the fundamental interplay between boundary value problems for ordinary differential equations and the topological structures they inhabit. Scholars and advanced students across geometry, topology, and dynamical systems will find in these pages a rich landscape of ideas and a fresh perspective on a classical subject.
