Chaos In Single And Double-Well Duffing Oscillators Under Certain Parametric Variations And Excitations.

                                                                        ABSTRACT

This study focuses on the generation and evolution of chaotic motions in single and double-well Duffing oscillators under certain parametrical excitations. The Melnikov approach and Lyapunov exponent are proposed to calculate the threshold values for the chaotic motion in a Duffing system. The minimum and maximum values were obtained and the dynamical behaviors showed the intersections of manifold which was illustrated with MATHCAD software. Similar results obtained from the two methods show that the behavior of the perturbed Duffing oscillator is chaotic and highly unstable with repeated resonances of successively higher periods. As a result, the functions with symmetric wells were separated by the barrier at a point and the unperturbed system produces three equilibria points showing similar behaviors. Also, the method of the Lyapunov exponents narrow the range for the critical threshold values and detect mutations in the chaotic system. Numerical simulations showed that as the parameter was varied, repeated resonances of successively higher periods occurred and unstable chaotic motion was observed.