Temporal and spectral responses of a softening Duffing oscillator undergoing route-to-chaos

Because nonlinear responses are oftentimes transient and consist of complex amplitude and frequency modulations, linearization would inevitably obscure the temporal transition attributable to the nonlinear terms, thus also making all inherent nonlinear effects inconspicuous. It is shown that linearization of a softening Duffing oscillator underestimates the variation of the frequency response, thereby concealing the underlying evolution from bifurcation to chaos. In addition, Fourier analysis falls short of capturing the time evolution of the route-to-chaos and also misinterprets the corresponding response with fictitious frequencies. Instantaneous frequency along with the empirical mode decomposition is adopted to unravel the multi-components that underlie the bifurcation-to-chaos transition, while retaining the physical features of each component. Through considering time and frequency responses simultaneously, a better understanding of the particular Duffing oscillator is achieved.