  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.
