A generalized harmonic function perturbation method for determining limit cycles and homoclinic orbits of Helmholtz—Duffing oscillator

In this paper, a novel description of periodic solution and homoclinic orbit of undamped Helmholtz—Duffing oscillator is proposed via nonlinear time transformation. Based on this novel description, a generalized harmonic function perturbation method is presented to determine the limit cycles and homoclinic orbits of Helmholtz—Duffing oscillator with nonlinear damping. The amplitude of limit cycle and critical value of the homoclinic bifurcation parameter can be also predicted. To illustrate the accuracy of the present method, the solutions obtained in this paper are compared with those of Runge—Kutta method, which shows the method proposed in this paper is effective and feasible.