A dissipative Benjamin–Ono equation is introduced to study turbulence of a magnetic two‐fluid system. An exact nonlinear mode truncation method is applied in which a finite number of poles are used to represent completely the asymptotic behavior of the system. The pole dynamics exhibit a wide range of nonlinear phenomena, including periodic and chaotic orbits. Statistical properties of the resultant turbulent flows such as correlation functions and energy spectra are also studied. The results show that the system can indeed reach the strong turbulent state.