Analytical and numerical solution to the nonlinear cubic Duffing equation: An application to electrical signal analysis of distribution lines

The Duffing oscillator represents an important model to describe mathematically the nonlinear behaviour of several phenomena occurring in physics and engineering. In this paper, analytical and numerical solutions to the nonlinear cubic Duffing equation governing the time behaviour of an electrical signal are found as a function of the magnitude and of the sign of the nonlinear parameter, of the damping parameter and for different values of the forcing term. A stability analysis of the Duffing equation in the absence of the forcing term is also performed as a function of the sign and magnitude of the nonlinear parameter. A fitting procedure of the Duffing solution to the current signal flowing in different distribution lines allows us to determine the degree of nonlinearity of the electrical signal suggesting a potential way to quantify the nonlinear behaviour of current electrical signals.