Bifurcation and chaos detection of a fractional Duffing–van der Pol oscillator with two periodic excitations and distributed time delay

This paper analytically and numerically investigates the dynamical characteristics of a fractional Duffing–van der Pol oscillator with two periodic excitations distributed time delay. First, we consider pitchfork bifurcation system driven by both high-frequency parametric excitation low-frequency external excitation. Utilizing method direct partition motion, original is transformed into an effective integer-order slow system, supercritical subcritical bifurcations are observed in this case. Then, study chaotic behavior when frequencies equal. The necessary condition for existence horseshoe chaos from homoclinic obtained based on Melnikov method. Besides, parameters effects routes to detected diagrams, largest Lyapunov exponents, phase portraits, Poincaré maps. It has been confirmed that theoretical predictions achieve high coincidence numerical results. techniques can be applied explore underlying dynamics fractional-order models.