  In this paper, we study a class of double phase systems which contain the
singular and mixed nonlinear terms. Unlike the single equation, the mixed
nonlinear terms make the problem more complicate. The geometry of the fibering
mapping has multiple possibilities. To overcome the difficulties posed by the
mixed nonlinear terms, we need to repeatedly construct concave functions,
discuss different cases, and use the properties of concave functions and basic
inequalities such as H\"{o}lder inequality, Poincar\'{e}'s inequality and
Young's inequality. By the use of the Nehari manifold, the existence and
multiplicity of positive solutions which have nonnegative energy are obtained.
