A perturbation-incremental method for the calculation of semi-stable limit cycles of strongly non-linear oscillators

Abstract

The semi-stable limit cycle and bifurcation of strongly non-linear oscillators of the form ẍ + g(x) = λƒ(x, ẋ, μ)ẋ is studied by the perturbation-incremental method. Firstly, the ordinary differential equation is transformed into an integral equation by a non-linear time transformation, then the initial solution for λ≈0 is obtained by using the perturbation method. Secondly, the solution for an arbitrary value of λ can be determined by using the incremental approach. Two examples are given to show the efficiency and accuracy of the present method.