Dynamical Response of a Van der Pol-Duffing System with an External Harmonic Excitation and Fractional Derivative

We examine the Van der Pol-Duffing system with external forcing and a memory possessing a fractional damping term. The system exhibits broad spectrum of nonlinear behavior including transitions from the periodic to nonperiodic motion. Replacing a first-order derivative damping term by a fractional damping one, we include to the system memory effect which increases the dimension of the dynamical system. As a consequence of such assumptions, the quantitative nonlinear analysis meets some limitations in this case. Instead of the well-known Lyapunov exponent treatment, we advocate to use the 0-1 test that combines both statistical and frequency properties of the attractor but does not depend on the dimension of the state space. The results have been confirmed by quantitative nonlinear analysis: bifurcation diagrams, phase portraits, Poincare sections, and the maximal Lyapunov exponent estimated in the limited two-dimensional phase space.