Suppression of chaos in a generalized Duffing oscillator with fractional-order deflection

A generalized Duffing oscillator with fractional-order deflection can be used to model the oscillatory motion of a buckled beam with simply supported or hinged ends. In this work, the problem of suppression of chaos in such a Duffing oscillator is considered. We show the appropriate range of parameters for the control of horseshoe chaos by introducing external periodic resonant excitation and parametric excitation as chaos-suppressing perturbation. Through the Melnikov technique, we obtain that in addition to the frequency, the phase difference between the chaos-inducing excitation and the chaos-suppressing excitation of systems plays a key role in chaos suppression. Given the optimum phase that satisfies the inhibition theorems, we compare the chaos-suppressing efficiency of external and parametric periodic perturbations for the principal resonance case. Compared with parametric (external) excitation, external (parametric) excitation with a frequency above (below) a critical value is more effective in suppressing homoclinic chaos because it provides a wider amplitude range. The results hold for an arbitrary deflection order as either an integer or a fraction, which depends on the material and bending properties of the beam, as long as its value is larger than 1. Moreover, the critical value of the frequency will shift to a larger value as the deflection order increases.