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

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.

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

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.