We propose a classification framework for relay systems in \(\mathbb {R}^n\) , distinguishing between generic phenomena and those of codimension one. We demonstrate the openness and density of these classifications, present unfoldings of normal forms, and analyze their dynamics to identify stable conditions within one-parameter families of these systems. We explore the local dynamics of relay systems in lower dimensions, particularly for \(n=2\) and \(n=3\) , focusing on those with codimension zero and one. Furthermore, for the \(n\) -dimensional case, we provide conditions under which relay systems are reversible and generic. In the two-dimensional case, we characterize when the origin is a center. In this context, we perturb this class of relay systems, breaking the symmetry with a perturbation parameter, and, using the Melnikov method, we prove that a repelling hyperbolic crossing limit cycle arises from the origin.
