The epsilon’s nonlinearity of permittivity and its effects upon the characteristics of a two-dimensional composite created by arrays of wires and split-ring resonators embedded into a nonlinear dielectric were reported. The dependences of the real part and imaginary part of the effective permittivity of the structure on the varying external magnetic field strength were calculated. It is demonstrated that the real part of the effective permittivity has hysteresis structures. This property of permittivity results in the switching of the composite among right-handed materials, left-handed materials, epsilon-negative materials, and mu-negative materials under different conditions, which can be controlled by a proper choice of the external magnetic-field strength (intensity of incident electromagnetic waves) in a certain frequency range. This phenomenon can be used to realize the micro-wave or optical switching devices.