Variational destruction of invariant circles

TL;DR

This paper demonstrates that by constructing specific generating functions close to integrable systems, one can eliminate invariant circles with a given rotation number in area-preserving twist maps, using variational methods.

Contribution

It introduces a variational approach to destroy invariant circles with prescribed rotation numbers in near-integrable twist maps, extending understanding of dynamical system stability.

Findings

Invariant circles can be destroyed for large n

Constructed generating functions are close to integrable systems

Method applies to arbitrary rotation numbers

Abstract

We construct a sequence of generating functions $(h_n)_{n\in\N}$, arbitrarily close to an integrable system in the $C^r$ topology with $r<4$ for $n$ large enough. With the variational method, we prove that for a given rotation number $\omega$ and $n$ large enough, the exact monotone area-preserving twist maps generated by $(h_n)_{n\in\N}$ admit no invariant circles with rotation number $\omega$.