Development of Chaos Diagrams for Duffing Oscillator Using Linearity and Nonlinearity Characteristics of Periodic and Chaotic Responses

摘要

This study exploited the computation accuracy of governing equations of linearly or periodically behaves dynamic system with fourth and fifth order Runge-Kutta algorithms to develop chaos diagrams of harmonically excited Duffing oscillator. The study adopt the fall to tolerance of absolute deviation between two independently sought solutions of governing equation to characterise excitation frequencies and amplitude parameter point of Duffing oscillator as either chaotic or not. Displacement and Velocity time history, Phase plot and Poincare were used to validate FORTRAN coded programmes used for this study and chaos diagrams developed at two different damping coefficients. The validation results agreed perfectly with those obtained in the literature. The chaos diagrams predicted by computation at two different damp coefficient levels conforms generally in trend to literature results by Dowell (1988) and qualitatively the same for three different combination of constant time based Runge-Kutta algorithms. The chances of chaotic behaviour of Duffing oscillator under the combined driven force of parameters becomes more than double at 0.0168 damping coefficient when compared with corresponding results at 0.168 damping coefficient. The probability that selected excitation frequencies and amplitudes will drive Duffing oscillator chaotically at 0.168 damp coefficients is 29.4%, 27.8% and 29.4% respectively. This study demonstrated the significant utility of numerical techniques in dealing with real-world problems that are dominantly nonlinear and shows that in addition to being sensitive to initial conditions, chaos is equally sensitivity to appropriate simulation time steps. In addition, the present chaos diagram generating numerical tool is uniquely characterised by being faster and predicting reliably than that earlier reported by the authors.