Limit Tori in Non-smooth Systems: an Analytic Detection Method and an Existence Result for Piecewise Linear Systems

Unlike linear vector fields, which exhibit simple dynamics, piecewise linear vector fields support significantly richer and more complex behavior. For instance, whereas planar linear vector fields are known to be devoid of limit cycles, their piecewise counterparts may exhibit limit cycles. In dimension three, although linear vector fields cannot support isolated invariant tori, it is natural to ask whether piecewise linear vector fields may admit such structures. In this context, the central objective of this paper is to establish the existence of both attracting and repelling invariant tori in 3D piecewise linear vector fields, which are referred to as limit tori. Motivated by this question, we develop a general analytical mechanism for detecting and characterizing isolated invariant tori in a broader class of piecewise smooth differential systems. This mechanism provides sufficient conditions for the existence of an invariant torus associated with a normally hyperbolic invariant closed curve of the first return map, arising through a Neimark–Sacker bifurcation. We use this mechanism to derive a family of 3D piecewise linear vector fields that exhibits isolated invariant tori, which are attracting or repelling depending on the parameter values and persist under small perturbations. To the best of our knowledge, this constitutes the first analytical detection of such attractors and repellers in piecewise linear vector fields.