Darboux integrability and algebraic invariant surfaces for the Rikitake system

Abstract

In this paper, we study the Darboux integrability of the Rikitake system ẋ=−μx+yz, ẏ=−μy+x(z−a), ż=1−xy. More precisely, we characterize all the invariant algebraic surfaces, the exponential factors, and the polynomial, rational, and Darboux first integrals of this system according to the values of its parameters a and μ.