Unstable periodic orbits in the intermediate Rayleigh-number regime

Construct unstable periodic orbits of two-dimensional Rayleigh–Bénard convection at Prandtl number Pr = 1 for Rayleigh numbers between approximately 6 × 10^6 and 1.4 × 10^7, where flow analysis indicates that such orbits may exist but the Newton–Krylov method did not identify them.

Background

The paper computes one steady state and three families of periodic orbits in two-dimensional Rayleigh–Bénard convection at Pr = 1. The identified branches organize the transitions from periodic dynamics through quasiperiodicity and phase locking toward chaos. However, between approximately Ra = 6 × 10^6 and the onset of chaos near Ra = 1.4 × 10^7, the direct numerical simulations appear to remain close to an unstable periodic orbit that could not be converged using the Newton–Krylov shooting-based procedure.

The authors attribute this failure to the difficulty of the Newton–Krylov method in finding longer-period orbits and identify the construction of invariant solutions without relying on a shooting strategy as a direction for future work. The unresolved problem is therefore to determine and compute the unstable periodic orbits organizing the dynamics in this intermediate regime.