Integrability of the Hide--Skeldon--Acheson dynamo

Abstract

In this work we consider the Hide-Skeldon-Acheson dynamo model \[ \dot x=x(y-1)-\beta z, \quad \dot y =\alpha(1-x^2)-\kappa y, \quad \dot z =x-\lambda z, \] where $\alpha,\beta,\kappa$ and $\lambda$ are parameters. We contribute to the understanding of its global dynamics, or more precisely, to the topological structure of its orbits by studying the integrability problem. Provided $\alpha \ne 0$ we identify the values of the parameters of this model, for which it admits a first integral. Also, as corollary of our main results we get that for $\alpha, \beta, \kappa \ne 0$ the dynamo model does not admit a polynomial, rational or Darboux first integral.