Planar constant piecewise smooth vector fields with large hysteresis

Mathematics > Dynamical Systems [Submitted on 18 Jun 2026] Title:Planar constant piecewise smooth vector fields with large hysteresis View PDF HTML (experimental)Abstract:Throughout this work, we will carry out a rigorous mathematical analysis of a class of control systems that is widely used in applications but still lacks a consistent theoretical foundation for describing the types of limit sets that may arise from its dynamics. There are applications in which, for example, a treatment for a given disease is administered until the level of diseased cells falls below a prescribed threshold C1. At that point, the treatment is suspended in order to allow the patient's organism to recover from its side effects. Subsequently, when the level of diseased cells reaches a second threshold C2 bigger than C1, the treatment is resumed, and the protocol is repeated. To the best of our knowledge, there is not a mathematical classification of such models. In this paper, we initiate what is intended to become a consistent body of literature aimed at determining the limit sets of such models. We begin with the planar case, in which two linear vector fields are active and two switching boundaries are considered. Naturally, in future developments, control systems in higher dimensions, featuring additional vector fields and more general switching manifolds, should also be considered. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.