Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der Pol-Duffing oscillator

Abstract

The intent of this work is to explore the generation mechanism of four mixed-mode vibration types triggered by the pitchfork bifurcation delay phenomenon in a driven van der Pol-Duffing oscillator. The four mixed-mode vibration types can be named “delayed sup-pitchfork/sup-Hopf” type for Homoclinic connection, symmetric “delayed sup-pitchfork/sup-Hopf” type, “point/point” type for delayed sup-pitchfork and “delayed sup-pitchfork/delayed Homoclinic” type, respectively. Our study proves that the pitchfork bifurcation delay phenomenon deserves a prominent role in the occurrence of the four mixed-mode vibration types, which enriches the dynamical behaviors of the bifurcation delay phenomenon as well as the possible routes to mixed-mode vibrations.