# Periodic orbits near a bifurcating slow manifold

Research output: Contribution to journalJournal articleResearchpeer-review

## Abstract

This paper studies a class of $1\frac12$-degree-of-freedom Hamiltonian systems with a slowly varying phase that unfolds a Hamiltonian pitchfork bifurcation. The main result of the paper is that there exists an order of $\ln^2\epsilon^{-1}$-many periodic orbits that all stay within an $\mathcal O(\epsilon^{1/3})$-distance from the union of the normally elliptic slow manifolds that occur as a result of the bifurcation. Here $\epsilon\ll 1$ measures the time scale separation. These periodic orbits are predominantly unstable. The proof is based on averaging of two blowup systems, allowing one to estimate the effect of the singularity, combined with results on asymptotics of the second Painleve equation. The stable orbits of smallest amplitude that are {persistently} obtained by these methods remain slightly further away from the slow manifold being distant by an order $\mathcal O(\epsilon^{1/3}\ln^{1/2}\ln \epsilon^{-1})$.
Original language English Journal of Differential Equations 259 9 4561–4614 0022-0396 https://doi.org/10.1016/j.jde.2015.06.006 Published - 2015

## Keywords

• Slow–fast systems
• Hamiltonian systems
• Separatrix crossing
• Normally elliptic slow manifolds

## Fingerprint

Dive into the research topics of 'Periodic orbits near a bifurcating slow manifold'. Together they form a unique fingerprint.