On the integrable stretch-twist-fold flow: Bi-Hamiltonian structures and global dynamics

Abstract

The stretch-twist-fold (STF) flow is a variant of the dynamo model describing the generation and behavior of magnetic fields in celestial bodies such as stars and planets. This study seeks to provide fresh insights into the integrable STF flow within the framework of dynamical systems theory and Poisson geometry. Our results include (i) the establishment of Poisson structures, Hamilton–Poisson realizations, and a Lax formulation for the STF flow; (ii) a comprehensive classification of phase portraits for the STF flow restricted to its symplectic leaf; (iii) a description of the asymptotic behavior of the STF flow on the Poincaré sphere, revealing the occurrence of bifurcations at infinity; (iv) a characterization of the energy-Casimir mapping of the STF flow and its connections with dynamical elements. These findings have the potential to deepen our understanding of the intricate and diverse dynamics exhibited by the STF flow in the context of dynamo theory.