Harmonic and Sub-Harmonic Periodic Solutions (1/2, 1/3) of Generalized Mathieu-Van der Pol-Duffing Equations

The frequency-locking area of harmonic and subharmonic ( 1/2, 1/3 ) solutions in a fast harmonic excitation Mathieu-Van der PolDuffing equation is studied. A perturbation technique is then performed on the slow dynamic near the harmonic and subharmonic ( 1/2, 1/3 ) solutions, to obtain reduced slow flow equations governing the modulation of amplitude and phase of the corresponding slow dynamics. Results show that fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. Numerical solutions are represented the analytical results.