Studies Of Localized Modes In Typical Anharmonic Discrete Systems

Harmonic approximation is a commonly used method.However,in realistic systems,the observed phenomena are often different from the results obtained by harmonic approximations,because there is anharmonic effect in both microscopic and macroscopic systems.As one of the most important effects in nonlinear physics,anharmonic effects can lead to huge changes of the physical properties of the systems.For a uniform lattice with periodic repeating unit structures,the anharmonic effect can cause self-localization of the system,which is similar to the localized mode formed by impurity in the simple harmonic lattice,often referred to as intrinsic localized mode or discrete breather;for lattices containing impurities,anharmonic effects also have an important impact on the physical properties of localized modes.The studies of localization in anharmonic discrete systems involve physics,engineering,mathematics and other important fields,and have great scientific value and potential application prospect.In the early years,the researchers of localization in anharmonic discrete systems mainly focused on the field of microscopic condensed matter,then the researchers gradually extend the research to micromechanical cantilever systems,electronic lattices,granular lattices,pendulum systems and so on.In this dissertation,three typical systems from small to large scale-microscopic condensed matter system,micromechanical cantilever system and electronic circuit system are selected,in which the localized mode is studied.Although the physical quantities that represent localization and the interaction mechanisms of these three systems are different,they can all be abstracted as anharmonic discrete models,and the properties of localized modes can be studied by similar mathematical methods.In the microscopic condensed matter system,intrinsic localized mode has been thoroughly studied,but the properties of anharmonic impurity mode and the quantitative relationship of localization still need to be further studied.In the micromechanical cantilever system,moving intrinsic localized modes become a hot topic after the study of stable intrinsic localized modes which are fixed in a certain position,and the factors affecting the moving directions of intrinsic localized modes are still under exploration.In the electronic circuit system,different electronic components,driving ways and relationships between units usually lead to the formation of different types of intrinsic localized modes,which are not fully explored yet.In view of these problems,some typical models are chosen for microscopic condensed matter system,micromechanical cantilever system and electronic circuit system,respectively,and the localized modes are studied.For the condensed matter system,rotating wave approximation is used to quantitatively describe energy localization as a function of the mass of impurity and anharmonic parameter in a simple one-dimensional anharmonic lattice.The bifurcation diagram is used to analyze the stability of symmetric and asymmetric localized modes.It is found that the central deviation of the localized mode exists in the one-dimensional anharmonic lattice containing a single impurity.In the two-dimensional anharmonic lattice,iterative analysis shows that light impurities and hard isobaric defects can always absorb part of the energy stably from the travelling anharmonic plane wave of longitudinal mode.In the micromechanical cantilever system,a one-dimensional cantilever array model with periodic boundary condition is utilized.Fourth order Runge-Kutta method is used to analyze the factors affecting the direction of the moving localized mode.It is found that the moving direction is not affected by the arrangement of the units,and is more susceptible to the driving way.The relationship between the quartic anharmonic interaction and the spatial scale of the localized modes is studied.It is found that in a certain frequency range,the driving frequency affects the direction of the moving localized mode.Under fixed boundary condition,the unstable moving localized modes tend to transfer its energy to the modes with frequency close to the frequency band,which makes the localized modes unstable.In the electronic lattice system,the intrinsic localized modes at Brillouin zone center and zone boundary are observed in one-dimensional electronic lattice with nearest and next nearest interactions.In the simple model without driving or damping,the resonance intrinsic localized mode excited at zone boundary gives its energy to the plane waves at the same frequency,which leads to instability of the intrinsic localized mode.In a more realistic model that includes driving and damping,numerical and experimental results both show that the resonant localized mode can be stabilized at the zone boundary by periodic driving.Whether excited at the zone boundary or at the zone center,stable intrinsic localized modes can be maintained by periodic driving.Moreover,the diatomic-like electronic lattice model is studied.It is found that in the band gap near the upper frequency branch,different kinds of localized modes are appearing successively as the frequency decreases.