On integrability of the segmented disc dynamo: the effect of mechanical friction

The segmented disc dynamos with or without friction are two basic models describing the self-excitation of a magnetic field, which are used to understand the generation of magnetic fields and the reversals in astrophysical bodies. This paper is devoted to giving a contribution to the understanding of their complexity by studying the integrability problem. (i) We first show that, generically, there exists a linear transformation to covert the segmented disc dynamo with friction (SDDF) into the Lorenz system, which yields that the integrability of SDDF can be obtained from the well-known results of the Lorenz system. (ii) The existence or non-existence of analytic, polynomial, rational, Darboux, C^1 first integrals for both segmented disc dynamo without friction (SDD) and SDDF are discussed. In particular, a complete classification of cases when SDD and SDDF admit a Darboux polynomial is provided. (iii) Non-integrability of SDD and SDDF is also discussed by the differential Galois method without assuming its closeness to an integrable system. To this end, a general non-integrability result is given, which can also be applied to study the non-integrability of other three-dimensional differential systems. (iv) We give a discussion about the effects of the friction on the dynamics for SDD and SDDF. Our results show that the friction term has a significant impact on both integrability properties and dynamic features of the segmented disc dynamo models, which may help us better understand the complex and rich dynamics of nonlinear homopolar disk dynamos.