A generic input-output analysis of zero-dispersion nonlinear resonance

– When a nonlinear oscillator is driven externally, it may become transparent, i.e., the output response appears without any dispersion or distortion. Such counter-intuitive feature, in fact, is a unique zero-dispersion nonlinear resonance (ZDNR). Herein, it is examined on the base of a generic input-output analysis. The conditions for ZDNR have been clarified and a power law relation has been established to feature the interplay with noise. The study of nonlinear oscillators has become the paradigm of nonlinear dynamics where a variety of new concepts, such as chaos and fractals, has been explored and established [1]. Its impact to physical research is significant since many systems, such as lasers [2], SQUID [3], relativistic oscillators [4], electrical circuits [5], polymeric molecules, and others, can be described by the models of this type. It is also remarkable to notice that nonlinear oscillators can be utilized to describe the dynamics of many physical systems with quantum-mechanical origins. For example, to mention a few, squeezing in classical oscillators [6]. On the other hand, in the classical aspect, resonance is a generic characteristic and the common feature is the dependence of the frequency of an eigenoscillator on its energy in which an extremum exhibits. Surprisingly, in the absence of dissipation, as shown by Soskin [7], a novel type of nonlinear resonance, zero-dispersion nonlinear resonance (ZDNR), can occur when a weak periodic force is applied for which the frequency is close to the external eigenfrequency. Since real systems are subject to dissipation, extension of ZDNR to a dissipative system has been an important issue. As further shown by Luchinsky, McClintock, Soskin, and Mannella, ZDNR can still be developed in dissipative systems [8]. Actually, ZDNR is very unique from a generic input-output point of view. A zero dispersion in spectral domain means that when the phases are fixed, in time domain the output can be exactly the same as the input, which suggests that transparent characteristics have been retrieved for the nonlinear oscillator. A similar feature of transparency also has been found in the interaction between light and matters, e.g., self-induced