# Time averages and periodic attractors at high Rayleigh number for Lorenz-like models

## TL;DR

This paper analyzes periodic solutions and time averages in Lorenz-like models at high Rayleigh numbers, proving existence and stability of symmetric periodic orbits, and exploring transport properties and hysteresis phenomena.

## Contribution

It provides a rigorous proof of symmetric periodic orbit existence and stability in Lorenz '63 equations at high Rayleigh numbers, expanding time averages using elliptic integrals.

## Key findings

- Existence and stability of symmetric periodic orbits confirmed.
- Transport can be arbitrarily small on periodic attractors.
- Hysteresis loop observed between periodic attractors and equilibria.

## Abstract

Revisiting the Lorenz '63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite $\rho$, we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average `transport', which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the non-zero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the non-zero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.