Noise-Induced Effects on Discrete Breathers in a Nonlinear Electrical Lattice

Abstract

A universal concern in lattice structures, whether they be naturally occurring or engineered, is exactly how energy can and does move within them. A significant phenomenon that has been shown theoretically, numerically, and experimentally to affect the behavior of energy in lattice structures is that of discrete breathers, also known as energy localizations or intrinsic localized modes. Discrete breathers affect the energy distribution in a lattice or array by concentrating it in localized and oscillatory fashion. While these have been known to occur in linear lattices with defects, they also occur in perfect (translationally invariant) nonlinear lattices of sufficient anharmonicity. This thesis seeks to further the study of the latter case, specifically by investigating how white and Lévy stable noise can be used to manipulate and create discrete breathers in a macro-scale nonlinear electric lattice. Using a dynamical model from literature and the Euler-Maruyama method of numerical integration, the effects of both additive white noise and Lévy stable noise on breathers in this array are investigated. These breathers are first initialized in the array using small randomized initial voltage conditions and a driving frequency below the system's natural frequency. It is found that additive white noise can in fact affect the number of breathers present by causing some of them to combine. Additionally, noise can be used to shorten the transient time before breather appearance. Importantly, to garner these effects, the noise must be applied during this transient phase. Applying it after will have little effect as already-formed breathers are too robust. The results are mostly the same for Lévy stable noise, although the large flights from the mean characteristic of this noise type require that the noise intensity used be reduced significantly. Next the ability of white noise to generate discrete breathers is investigated. The initial conditions are set to zero, meaning that no breathers will form in the system without some kind of intervention. Applying noise for either the entire simulation duration or only the first half both resulted in the consistent and reliable creation of discrete breathers. Additionally, it is found that a temporary burst of noise across the array can be used to create breathers on command at any time. The breathers created will exhibit some noisy behavior, but this can be minimized by applying the necessary noise to only one cell. Thus, additive noise can serve as a useful means of breather creation, a key result of this thesis. These results testify to the significance of the interactions between the discrete breathers in this nonlinear electric line and noise. Noise can influence the number of breathers present, and even create breathers on command. The latter result is of particular significance and represents a promising area for future work.