Analytical solution of the Duffing equation

Abstract

Purpose

The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.

Design/methodology/approach

The double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.

Findings

Expressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.

Originality/value

The results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.

Citation

Georgiev, Z., Trushev, I., Todorov, T. and Uzunov, I. (2021), "Analytical solution of the Duffing equation", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 40 No. 2, pp. 109-125. https://doi.org/10.1108/COMPEL-10-2019-0406