Braid invariants and instability of periodic solutions of time-periodic $2$-dimensional ODE's

Abstract

We present a topological approach to the problem of the existence of unstable periodic solutions for 2-dimensional, time-periodic ordinary differential equations. This approach makes use of the braid invariant, which is one of the topological invariants for periodic solutions exploiting a concept in the low-dimensional topology. Using the braid invariant, an equivalence relation on the set of periodic solutions is defined. We prove that any equivalence class consisting of at least two solutions must contain an unstable one, except one particular equivalence class. Also, it is shown that more than half of the equivalence classes contain unstable solutions.