The scope of this work are electric circuits or electronic devices with chaotic regimes, in particular the Chua’s circuit. A nonlinear analysis of chaotic attractors based on the Krot’s method of matrix decomposition of vector functions in state-space of complex systems has been used to investigate the Chua’s circuit with smooth nonlinearity. It includes an analysis of linear term of the matrix series as well as an estimation of influence of high order terms of this series on stability of complex system under investigation. Here the method of matrix decomposition has been applied to analysis of the Chua’s attractor. The terms of matrix series have been used to create a simulation model and to reconstruct an attractor of chaotic modes. The proposed simulation model makes it possible to separate an influence of nonlinearities on forming a chaotic regime of the Chua’s circuit. Usage of both the matrix decomposition method and computational experiment has allowed us to find out that the initial turbulence model proposed by L. D. Landau is suitable for set-up description of the chaotic regime of the Chua’s circuit. It is shown that a mode of hard self-excitation in the Chua’s circuit leads to its chaotic regime operating with a double-scroll attractor in the state-space. The results might be used to generate of chaotic oscillations or data encryption.