Investigation of Poincare Solutions of Nonlinear Duffing and Pendulum under Selected Periodic Excitations Using Fractal Disk Characterisation

Abstract

Literature has shown that harmonically excited nonlinear Duffing and pendulum oscillators can respond chaotically under the influence of some of their drive parameters combination. However, literature is scarce on the steady state responses of these oscillators when excited arbitrarily and periodically. Therefore, this research was designed to investigate the potential qualitative and quantitative variation in the steady Poincare solutions of nonlinear Duffing and pendulum oscillators under selected periodic excitations compared to their harmonically excited counterparts. The non-dimensional second Order Differential Equation (ODE) corresponding respectively to governing equations for harmonically/periodically excited nonlinear Duffing and pendulum were solved using the constant step fourth order Runge-Kutta algorithms. The corresponding steady state Poincare solutions obtained were characterised by visual inspection and fractal dimension measure obtained using fractal disk counting method. Visual inspection of corresponding steady Poincare solutions show that they are qualitatively indistinguishable. However, the corresponding estimated fractal dimension varied significantly. The absolute variation in dimension was found to be between 1.37% and 4.92% for the Duffing oscillator and between 5.67% and 7.39% for the pendulum oscillator.