Electromagnetic Oscillations in a Driven Nonlinear Resonator: A New Description of Complex Nonlinear Dynamics

TL;DR

This paper presents an exact solution for electromagnetic oscillations in a driven nonlinear resonator, revealing complex dynamics including singular continuous spectra, which are significant for understanding nonlinear systems and chaos.

Contribution

It provides the first exact integration of Maxwell equations in a nonlinear cavity with exponential dielectric response, demonstrating fractal spectra in an exactly solvable system.

Findings

Electromagnetic oscillations exhibit complex temporal behavior.

Spectra contain singular continuous (fractal) components.

First demonstration of fractal spectra in an exactly integrable nonlinear system.

Abstract

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. To the best of our knowledge, this is the first demonstration of the existence of a singular continuous (fractal) spectrum in an exactly integrable system.