Comparative Analysis of Numerically Computed Chaos Diagrams in Duffing Oscillator

Abstract

This study utilised optimum fractal disk dimension algorithms to characterize the evolved strange attractor (Poincare section) when adaptive time steps Runge-Kutta fourth and fifth order algorithms are employed to compute simultaneously multiple trajectories of a harmonically excited Duffing oscillator from very close initial conditions. The challenges of insufficient literature that explore chaos diagrams as visual aids in dynamics characterization strongly motivate this study. The object of this study is to enable visual comparison of the chaos diagrams in the excitation amplitude versus frequency plane. The chaos diagrams obtained at two different damp coefficient levels conforms generally in trend to literature results[1] and qualitatively the same for all algorithms. The chances of chaotic behaviour are higher for combined higher excitation frequencies and amplitudes in addition to smaller damp coefficient. Fourth and fifth order Runge-Kutta algorithms indicates respectively 62.3% and 53.3% probability of chaotic behaviour at 0.168 damp coefficient and respectively 77.9% and 78.9% at 0.0168 damp coefficient. The chaos diagrams obtained by fourth order algorithms is accepted to be more reliable than its fifth order counterpart, its utility as tool for searching possible regions of parameter space where chaotic behaviour/motion exist may require additional dynamic behaviour tests.