# Equilibria, periodic orbits and computing them

## TL;DR

This paper explains how to find equilibria and periodic orbits using the flow map and Jacobian-free Newton-Krylov method, providing practical code examples for complex systems like pipe flow and the Lorenz system.

## Contribution

It introduces a Jacobian-free Newton-Krylov approach for computing equilibria and periodic orbits that is easy to implement with existing time stepping codes.

## Key findings

- Method successfully applied to Lorenz system
- Code is problem-independent and scalable to large systems
- Enables efficient computation of complex dynamical structures

## Abstract

In this short exposition, we describe equilibria and periodic orbits in terms of the flow map, {\Phi}, and discuss the essentials of the Jacobian-free Newton-Krylov (JFNK) method that can be used to find them. This method requires little more than calls to an existing time stepping code, which {\Phi} can be considered to represent. Fortran90 / MATLAB code is available to try it out for yourself, where, in the template/example the method is applied to the Lorenz system. This code is problem-independent and can be applied to large systems, having initially been developed to find periodic orbits in simulations of pipe flow.