A nonlinear optimization bifurcation tracking method for periodic solution of nonlinear systems

A continuation strategy which exploits the proposed optimization formulation to search periodic solution is presented for bifurcation tracking. The proposed optimization correction scheme integrated with the prediction-correction strategy is characterized by the minimization of the pseudo arclength equation with the imposition of optimization constraints on the harmonic balance equations and bifurcation conditions. Stability and sensitivity analysis are carried out in the frequency domain by making use of the Floquet theory. The derivatives of the stability factor with respect to the Fourier coefficients, vibration frequency, and bifurcation parameters are given in the explicit form. Finally, the effectiveness of the proposed methodology is illustrated by three numerical examples which include a Duffing oscillator, a nonlinear energy sink, and a Jeffcott rotor, respectively. Numerical results have demonstrated that the proposed method offers a convenient scheme to trace bifurcation solution.