Approximate solutions of a class of complex nonlinear dynamical systems

Nonlinear dynamical systems, being a realistic representation of nature, often exhibit a somewhat complicated behaviour. Their analysis requires a thorough investigation into the solutions of the governing nonlinear differential equations. In this paper, an approximate method is presented for solving complex nonlinear differential equations of the form: z̈+ω2z+εf(z,z̄,ż,z̄̇)=0,where z is a complex function and ε is a small parameter. It is based on the generalized averaging method which we have developed recently. Our approach can be viewed as a generalization of the approximate method based on the Krylov–Bogoliubov averaging method. The study of these systems is of interest to several fields of statistical mechanics, physics, electronics and engineering. Application of this method to special cases is performed for the purpose of comparison with numerical computations. Excellent agreement is found for reasonably large values of ε, which shows the applicability of this method to this kind of nonlinear dynamical systems. This agreement gives extra confidence that the analytical results are correct. These analytical results can be used as a theoretical guidance for doing further numerical or theoretical studies.