A computer-assisted proof of dynamo growth in the stretch-fold-shear map

Abstract

The Stretch-Fold-Shear (SFS) operator Sα is a functional linear operator acting on complex-valued functions of a real variable x on some domain containing [−1,1] in R. It arises from a stylized model in kinematic dynamo theory where magnetic field growth corresponds to an eigenvalue of modulus greater than 1. When the shear parameter α is zero, the spectrum of Sα can be determined exactly, and the eigenfunctions corresponding to non-zero eigenvalues are related to the Bernoulli polynomials. The spectrum for α > 0 has not been rigorously determined although the spectrum has been approximated numerically. In this paper, a computer-assisted proof is presented to provide rigorous bounds on the leading eigenvalue for α ∈ [0,5], showing inter alia that Sα has an eigenvalue of modulus greater than 1 for all α satisfying π/2 < α ≤ 5, thereby partially confirming an outstanding conjecture on the SFS operator.