Regular and Chaotic Vibrations of Van Der Pol and Rayleigh Oscillators Driven by Parametric Excitation

Interactions between two parametrically coupled self-excited oscillators are analysed in the paper. Self-excitation of each oscillator is defined by different functions, van der Pol's or Rayleigh's, while stiffness is assumed as a nonlinear Duffing's type term. The oscillators are coupled by a spring with periodically changing in time stiffness. Regular vibrations of the system, characteristic bifurcation points and the synchronisation regions are investigated by the analytical multiple scale of time method near the frequency locking zones corresponding to the principal parametric resonance. The results are verified numerically. Regions of possible chaotic motion are determined by direct numerical simulations. The Lyapunov's exponent criterion is used for classification of different types of motion.