TY - JOUR AU - Matsuoka, Takashi PY - 1999/12/01 Y2 - 2024/03/29 TI - Braid invariants and instability of periodic solutions of time-periodic $2$-dimensional ODE's JF - Topological Methods in Nonlinear Analysis JA - TMNA VL - 14 IS - 2 SE - DO - UR - https://apcz.umk.pl/TMNA/article/view/TMNA.1999.033 SP - 261 - 274 AB - We present a topological approach to the problem of the existence of unstable periodic solutionsfor 2-dimensional, time-periodic ordinary differential equations. This approach makes use of the braid invariant, which isone of the topological invariants for periodic solutions exploiting a concept in the low-dimensional topology.Using the braid invariant, an equivalence relationon the set of periodic solutions is defined.We prove that any equivalence class consisting of at least twosolutions 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. ER -