A computer-assisted proof of dynamo growth in the stretch-fold-shear map. Issue 1 (2nd January 2023)

• Record Type: Journal Article
• Authors: Pramy, F. A.; Mestel, B. D.; Gilbert, A. D.
• Is Part Of: Dynamical systems. Volume 38:Issue 1(2023)
• Page Start: 102
• Page End: 120
• Publication Date: 2023-01-02
• Subjects: Kinematic dynamo -- stretch-fold-shear map -- operator spectrum -- computer-assisted proof; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2022.2139224
• Languages: English
• ISSNs: 1468-9367

Abstract

Abstract : The Stretch-Fold-Shear (SFS) operator S α 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 S α 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 S α 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.