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arXiv · 1906.03528

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

Matthew D. Kvalheim·Anthony M. Bloch
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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.

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Matthew D. Kvalheim, Anthony M. Bloch. 2019-06-08. Families of periodic orbits: closed 1-forms and global continuability. https://arxiv.org/abs/1906.03528

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