@article{Matsuoka_1999, title={Braid invariants and instability of periodic solutions of time-periodic $2$-dimensional ODE’s}, volume={14}, url={https://apcz.umk.pl/TMNA/article/view/TMNA.1999.033}, abstractNote={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.}, number={2}, journal={Topological Methods in Nonlinear Analysis}, author={Matsuoka, Takashi}, year={1999}, month={Dec.}, pages={261–274} }