Resonance, symmetry, and bifurcation of periodic orbits in perturbed Rayleigh-B\'enard convection

TL;DR

This paper explores the global structure, resonance, symmetry, and bifurcation of periodic orbits in a perturbed Rayleigh-Bénard convection model, revealing symmetric properties and bifurcation behaviors as perturbation amplitude varies.

Contribution

It provides a detailed numerical analysis of the global structures and bifurcations of periodic orbits, highlighting symmetry properties and resonance phenomena in a perturbed Hamiltonian convection model.

Findings

Existence of symmetric properties of resonant periodic orbits.

Identification of bifurcation structures as perturbation amplitude varies.

Global phase space structures of periodic orbits in Rayleigh-Bénard convection.

Abstract

This paper investigates the global structures of periodic orbits that appear in Rayleigh-B\'enard convection, which is modeled by a two-dimensional perturbed Hamiltonian model, by focusing upon resonance, symmetry and bifurcation of the periodic orbits. First, we show the global structures of periodic orbits in the extended phase space by numerically detecting the associated periodic points on the Poincar\'e section. Then, we illustrate how resonant periodic orbits appear and specifically clarify that there exist some symmetric properties of such resonant periodic orbits which are projected on the phase space; namely, the period $m$ and the winding number $n$ become odd when an $m$-periodic orbit is symmetric with respect to the horizontal and vertical center lines of a cell. Furthermore, the global structures of bifurcations of periodic orbits are depicted when the amplitude $\varepsilon$ of the perturbation is varied, since in experiments the amplitude of the oscillation of the convection gradually increases when the Rayleigh number is raised.