Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

• More general forms of damping in terms of fractional derivative and time delay are investigated.
• The damping characteristics in Duffing–van der Pol oscillators are taken as examples.
• The steady state limit cycle bifurcations are solved by the residue harmonic balance.
• Rapid convergence is obtained by solving linear ODE.
• Qualitative differences in the damping characteristics for these dampers are found for the first time.

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.