SANTOS, D. M. G.; http://lattes.cnpq.br/9502411491585338; SANTOS, Desiane Maiara Gomes dos.
Resumo:
In this dissertation, we studied the dynamics of parametrically-driven oscillators, with a focus on the amplification of small signals. We begin with a revision of parametric resonance and parametric amplification in a linear oscillator parametrically excited. Next,
we studied two types of nonlinear parametrically-driven oscillators and finished the dissertation with an analysis of a parametric dimer. Basically, we analyzed the phenomena
of parametric resonance and parametric amplification by comparing the results obtained
analytically (via the averaging or harmonic balance methods) with those of numerical
integration of the equations of motion. In all cases, we obtained the transition line to
parametric instability of the parametric oscillator. We excited the parametric amplifier
with and without detuning between the pump and the external signal. We found that the
parametric amplification depends sensitively on the phase of the external ac signal and on
the internal pump amplitude. We showed that such amplifiers can be easily used for the
reception and decoding of signals with phase modulation. Furthermore, we obtained time
series, envelopes, and Fourier transforms of the response of the parametric amplifier to
small external ac signals. Specifically in the cases of the parametrically-driven Duffing oscillators, we obtained and analysed the bifurcation lines and the amplitude of limit cycles as function of the pump amplitude and frequency. In addition, we derived an expression for the signal and idler gains of the nonlinear parametrically-driven oscillators with the harmonic balance method. The results imply that the nonlinear parametric amplifiers can be excellent detectors, specially near bifurcations to instability, due to their high gains and narrow bandwidths. Finally, we studied the dynamics of two linear oscillators coupled and parametrically excited, with and without external ac driving. We found that such systems have a wealth of dynamical responses. They present parametric amplification that is dependent on the coupling parameter and on the phases of the external ac signals. Such systems may be used as tunable amplifiers.