טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentPerchikov Nathan
SubjectNonlinear Dynamics of Discrete Mechanical Systems with Flat
Dispersion Bands
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Oleg Gendelman


Abstract

Physical lattices with configurational local symmetries possess interesting features, an apparent one being the existence of flat dispersion curves (bands) in the linear spectrum. Such flat bands may be associated with the existence of hidden dynamic modes, which cannot be excited externally in the linear regime. An additional phenomenon associated with the flat bands is the emergence of detached, spatially localized, perfectly compact dynamic modes, which can have non-trivial stability characteristics in the nonlinear case. Many examples of lattices with flat bands were discussed in the recent literature, among others photonic lattices with Kerr nonlinearity, chains of coupled pendula, granular systems with Hertzian contact, Bose-Einstein condensates, and DNA molecules. One interesting example of a nonlinear lattice with a flat band is a discrete mechanical system of light linearly connected 'boxes' with two identical internal oscillators in each, which exhibit anharmonicity at large amplitudes. Analysis of nonlinear normal modes in such a system requires development of novel computational approaches.


Another system with an emerging flat dispersion band is a system of two chains of linearly connected masses with cross-linking to nearest neighbors, where, in addition, there are displacement-limiters producing impacts. Piecewise-linear systems of this sort have a noteworthy advantage. The stability of periodic solutions in piecewise-linear systems can be analyzed with high accuracy and efficiency using analytic construction of the monodromy matrix, which simplifies the study of local bifurcations. On the other hand, the impacts render such systems strongly nonlinear, which makes the study of global bifurcations by averaging techniques, such as canonic transformations, much harder. Recently introduced theoretical approaches, such as the application of the Action-Angle formalism in conjunction with the notion of the Limiting Phase Trajectory enable the execution of analysis of global bifurcations of small non-integrable piecewise-linear systems.


This PhD work presents results concerning the mentioned concepts of: hidden modes in smoothly nonlinear systems, including the development of a suitable local stability analysis method; compact modes, including analytic local stability results for a piecewise-linear non-smooth system; and global analysis and prediction of critical transitions in a strongly nonlinear (non-smooth) case, using a generalization of the concept of the Limiting Phase Trajectory.