Newton flows for elliptic functions: A pilot study

Elliptic Newton flows are generated by a continuous, desingularized Newton method for doubly periodic meromorphic functions on the complex plane. In the special case, where the functions underlying these elliptic Newton flows are of second-order, we introduce various, closely related, concepts of structural stability. In particular, within the class of all (second order) elliptic Newton flows, structural stability turns out to be a generic property. Moreover, the phase portraits of all such structurally stable flows are equal, up to conjugacy. As an illustration, we treat the elliptic Newton flows for the Jacobi functions sn, cn and dn.