Initial conditions-independent limit cycles of a fractional-order van der Pol oscillator

Abstract

The fractional-order van der Pol (vdP) oscillator, which is a typical self-excited system containing fractional derivatives, has increasingly stimulated the interest of many researchers. To the best of our knowledge, the existence and uniqueness of its limit cycle (LC) remains an open question. The major purpose of this study is to reveal the initial conditions independence of the LC. In order to solve the considered system accurately and efficiently, we propose a modified predictor–corrector numerical algorithm and a semi-analytical approach based on the homotopy analysis method. Numerical examples show that, similar to the integer-order vdP oscillator, the phase trajectories starting with different initial conditions converge to the same LC as the solution domain is long enough. It is also found that the convergence rate strongly depends on the initial conditions. On some occasions, the transient responses approach the steady state so slowly that it is possible to come to the misjudgment that there are a series of LCs depending on initial conditions.