Ignoring nonlinear effects in many practical situations degrades the performance. This paper considers nonlinear system characterization using higher-order cumulants and polyspectra. A novel method to blindly identify the kernels of cubic systems using mixed third- and fourth-order cumulants is developed. We study the link between the Fourier transform of third-order and fourth-order cumulants in nonlinear cubic systems. Then, we use the inverse Fourier transform to build a new formula which combines third- and fourth-order cumulants. Then, we generalize it to the nth-order cumulants and the kernels of nonlinear cubic systems driven by a non-Gaussian random signal, independent, identically distributed (i.i.d.) in Gaussian noise environment. Our performances results indicate that the proposed approach is able to identify blindly the kernels in cubic systems.