Optimization analysis of Duffing oscillator with fractional derivatives

The improved version of the constrained optimization harmonic balance method is presented to solve the Duffing oscillator with two kinds of fractional order derivative terms. The analytical gradients of the objective function and nonlinear quality constraints with respect to optimization variables are formulated and the sensitivity information of the Fourier coefficients can also obtained. A new stability analysis method based on the analytical formulation of the nonlinear equality constraints is presented for the nonlinear system with fractional order derivatives. Furthermore, the robust stability boundary of periodic solution can be determined by the interval eigenvalue problem. In addition, the sensitivity information mixed with the interval analysis method is used to quantify the response bounds of periodic solution. Numerical examples show that the proposed approach is valid and effective for analyzing fractional derivative nonlinear system in the presence of uncertainties. It is illustrated that the bifurcation solution in the fractional nonlinear systems may not be sensitive to the variation of the influence parameters.