Lucid Analysis of Periodically Forced Nonlinear Systems via Normal Forms

This paper presents a straightforward methodology to analyze periodically forced nonlinear systems with constant and periodic coefficients via normal forms. We demonstrate how the intuitive system state augmentation facilitates construction of normal forms by avoiding ad-hoc addition of equation variables, book-keeping parameters and detuning parameters. Moreover, this technique directly connects the periodic forcing terms and periodic coefficients of the nonlinearity with the augmented states — making it applicable to all periodically forced nonlinear systems. Accuracy of this approach is successfully verified via fulfilled compliance between analytical and numerical results of forced Duffing’s equation and Mathieu-Duffing equation.

FRACTIONAL DUFFING'S EQUATION AND GEOMETRICAL RESONANCE

We investigate the Fractional Duffing equation in the presence of nonharmonic external perturbations. We have applied the concept of Geometrical Resonance to this equation. We have obtained the conditions that should be satisfied by the external driving forces in order to produce high-amplitude periodic oscillations avoiding chaos. We also show that, for Duffing's equation with fractional damping, the perturbations that satisfy the Geometrical Resonance conditions are nonperiodic functions.

Ninety years of Duffing’s equation

In the paper the origin of the so named ?Duffing?s equation? is shown. The author?s generalization of the equation, her published papers dealing with Duffing?s equation and some of the solution methods are presented. Three characteristic approximate solution procedures based on the exact solution of the strong cubic Duffing?s equation are shown. Using the Jacobi elliptic functions the elliptic-Krylov-Bogolubov (EKB), the homotopy perturbation and the elliptic-Galerkin (EG) methods are described. The methods are compared. The advantages and the disadvantages of the methods are discussed.