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

Takashi Matsuoka

DOI: http://dx.doi.org/10.12775/TMNA.1999.033

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.

Keywords


Time-periodic 2-dimensional ODE; braid; unstable periodic solution

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