# Families of periodic orbits: closed 1-forms and global continuability

## TL;DR

This paper develops a new theoretical framework for the global continuation of periodic orbits in differential equations using closed 1-forms, enabling existence proofs without trapping regions or period bounds.

## Contribution

It introduces a novel notion of global continuability based on closed 1-forms and extends existing continuation theory to this setting, allowing broader existence proofs.

## Key findings

- New global continuation theorem for periodic orbits using closed 1-forms
- Ability to prove existence of periodic orbits without trapping regions
- Applicability demonstrated in synthetic biology-inspired examples

## Abstract

We investigate global continuation of periodic orbits of a differential equation depending on a parameter, assuming that a closed 1-form satisfying certain properties exists. We begin by extending the global continuation theory of Alexander, Alligood, Mallet-Paret, Yorke, and others to this situation, formulating a new notion of global continuability and a new global continuation theorem tailored for this situation. In particular, we show that the existence of such a 1-form ensures that local continuability of periodic orbits implies global continuability. Using our general theory, we then develop continuation-based techniques for proving the existence of periodic orbits. In contrast to previous work, a key feature of our results is that existence of periodic orbits can be proven (i) without finding trapping regions for the dynamics and (ii) without establishing a priori upper bounds on the periods of orbits. We illustrate the theory in examples inspired by the synthetic biology literature.