Fold-Hopf bifurcation, steady state, self-oscillating and chaotic behavior in an electromechanical transducer with nonlinear control

We investigate the nonlinear dynamics of an electromechanical transducer formed by a ferromagnetic mobile piece subjected to harmonic base oscillations. The normal form theory is applied to analyze the stability conditions of a Fold-Hopf bifurcation generated by a nonlinear control law consisting of an excitation electric tension and an external force applied to the mobile piece. The self-oscillating behavior is studied from the Krylov–Bogoliuvov method to deduce an approximate equation for the frequency of the oscillation that arises from the Fold-Hopf bifurcation. This information is used to choose appropriate values for the amplitude and frequency of the harmonic base vibrations to obtain chaotic oscillations. The chaotic motions are examined by means of sensitivity dependence, Lyapunov exponents, power spectral density, Poincaré sections and bifurcation diagram. The chaotic oscillations are used in conjunction with the nonlinear control law to drive the mobile piece to a desired set point. A detailed computational study allows us to corroborate the analytical results.