TY - JOUR
AU - Matsuoka, Takashi
PY - 1999/12/01
Y2 - 2023/04/01
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 -