INTRODUCTION

An interesting asymptotic limit in the theory of dynamical systems enforces a highly-developed chaotic structure by the assumption of large-amplitude particle excursion in flows and maps. An example of such a method applied to diffusion of a scalar is given by Rechester & White (1980). This important idea has been developed by Soward (1992) in the context of fast dynamo theory and in particular for the case of pulsed helical waves. Our purpose in this note is to apply the large-amplitude method of Soward (1992), to the simpler SFS map (Bayly & Childress 1987, 1988). In the SFS (stretchfold- shear) map, a simple baker's map in the xy-plane is supplemented by a lateral shear in the z-direction. Numerical calculations indicate that, when the map operates in a perfectly conducting fluid on a magnetic field of the form (B(y)eikz, 0, 0), the average of the field over planes z=constant can be made to grow exponentially for sufficiently large shear. This property of ‘perfect’ fast dynamo action has never been proved in the SFS problem, however, despite the existence of an especially simple adjoint eigenvalue problem, where the growing eigenfunctions, if they exist, are known to be smooth (Bayly & Childress 1988). Moreover, numerical studies show clearly the existence of these eigenfunctions for the perfect fast dynamo problem.