Solution to a Damped Duffing Equation Using He’s Frequency Approach

Abstract

In this paper, we generalize He’s frequency approach for solving the damped Duffing equation by introducing a time varying amplitude. We also solve this equation by means of the homotopy method and the Lindstedt–Poincaré method. High accurate formulas for approximating the Jacobi elliptic function cn are formally derived using Chebyshev and Pade approximation techniques.

Modified Homotopy Perturbation Method Based Solution of Linearly Damped Duffing Oscillator and Comparison With Simulated Solution*

“…The Duffing oscillator has been used to explain many observed phenomena in science, engineering, biological systems in particular nano-tubes, microtubules and hence, dynamical analysis of this oscillator attracted many workers [1,3,10,13,15]. Numerous researchers contributed to both analytical and numerical solutions of the Duffing oscillator with and without damping [4,16,17,18]. In a similar way the problem related to synchronization of chaotic Duffing system has also been taken up in recent years in [1,15].…”

“…Duffing equation without a damping term represents a conservative system. In view of the nonlinear characteristic of the basic Duffing oscillator, several authors have developed different analytical methods to obtain approximate analytical solution so as to understand the complexity of the involved dynamics [10,15,18]. Interestingly, the solution of Duffing oscillator, in case of non-conservative system, involves intricacies that led to several methods for the situation when damping coefficient is large [10,12].…”