A topological degree theory for rotating solutions of planar systems

TL;DR

This paper introduces a generalized topological degree for rotating solutions in planar systems, establishing a twist theorem for periodic solutions and applying it to various differential equations.

Contribution

It develops a new degree concept for rotating solutions and proves a twist theorem, extending classical results like the Poincaré-Birkhoff theorem.

Findings

Established a relation between the new degree and Brouwer's degree

Proved a twist theorem for periodic solutions in planar systems

Applied the theory to asymptotically linear and superlinear differential equations

Abstract

We present a generalized notion of degree for rotating solutions of planar systems. We prove a formula for the relation of such degree with the classical use of Brouwer's degree and obtain a twist theorem for the existence of periodic solutions, which is complementary to the Poincar\'e-Birkhoff Theorem. Some applications to asymptotically linear and superlinear differential equations are discussed.