Electromagnetic oscillations in a driven nonlinear resonator: a description of complex nonlinear dynamics

Many intriguing properties of driven nonlinear resonators, including the appearance of chaos, are very important for understanding the universal features of nonlinear dynamical systems and can have great practical significance. We consider a cylindrical cavity resonator driven by an alternating voltage and filled with a nonlinear nondispersive medium. It is assumed that the medium lacks a center of inversion and the dependence of the electric displacement on the electric field can be approximated by an exponential function. We show that the Maxwell equations are integrated exactly in this case and the field components in the cavity are represented in terms of implicit functions of special form. The driven electromagnetic oscillations in the cavity are found to display very interesting temporal behavior and their Fourier spectra contain singular continuous components. This is a demonstration of the existence of a singular continuous (fractal) spectrum in an exactly integrable system.