Detecting Limit Tori in Non-Smooth Systems: An Analytic Approach with Applications to 3D Piecewise Linear Systems

TL;DR

This paper develops an analytical method to detect invariant tori in non-smooth, piecewise linear 3D systems, demonstrating the existence and persistence of such structures under perturbations, a novel achievement in the field.

Contribution

It introduces a new analytical approach for identifying invariant tori in piecewise linear systems, including conditions for bifurcations and examples of persistent limit tori.

Findings

Derived conditions for Neimark--Sacker bifurcation in piecewise maps

Constructed 3D piecewise linear systems with attracting and repelling tori

First analytical detection of limit tori in piecewise linear systems

Abstract

This work investigates a class of non-autonomous $T$-periodic piecewise smooth differential systems and their associated time-$T$ maps. Our main result provides an analytical approach for detecting, within this class of piecewise differential systems, isolated invariant tori associated with normally hyperbolic invariant closed curves of the time-$T$ map. To achieve this, we derive sufficient conditions under which smooth near-identity maps undergo a Neimark--Sacker bifurcation. As an application of our main result, we present a family of 3D piecewise linear differential systems exhibiting attracting and repelling isolated invariant tori which, moreover, persist under small perturbations. To the best of our knowledge, this family provides the first examples in which limit tori are analytically detected in piecewise linear systems.