BIFURCATION AND CHAOS OF NONLINEAR VIBRO-IMPACT SYSTEMS

Vibro-impact systems are extensively used in engineering and physics field, such as impact damper, particle accelerator, etc. These systems are most basic elements of many real world applications such as cars and aircrafts. Such vibro-impact systems possess both the continuous characteristics as continuous dynamical systems and discrete characteristics introduced by impacts at the same time. Thus, an appropriately developed discrete mapping system is required for such vibro-impact systems in order to simplify investigation on the complexity of motions. In this dissertation, a few vibro-impact oscillators will be investigated using discrete maps in order to understand the dynamics of vibro-impact systems. Before discussing the nonlinear dynamical phenomena and behaviors of these vibro-impact oscillators, the theory for nonlinear discrete systems will be applied to investigate a two-dimensional discrete system (Henon Map). And the complete dynamics of such a nonlinear discrete dynamical system will be presented using the inversed mapping method. Neimark bifurcations in such a discrete system have also drawn a lot of interest to the author. The Neimark bifurcations in such a system have actually formed a boundary dividing the stable solution of positive and negative maps (inversed mapping). For the first time, one is able to obtain a complete prediction of both stable and unstable solutions in such a discrete dynamical system. And a detailed parameter map will be presented to illustrate how changes of parameters could affect the different solutions in such a system. Then, the theory of discontinuous dynamical systems will be adopted to investigate the vibro-impact dynamics in several vibro-impact systems. First, the bouncing ball dynamics will be analytically discussed using a single discrete map. Different types of motions (periodic and chaotic) will be presented to understand the complex behavior of this simple model. Analytical condition will be expressed using switching phase of the system in order to easily predict stick and grazing motion. After that, a horizontal impact damper model will be studied to show how complex periodic motions could be developed analytically. Complete set of symmetric and asymmetric periodic motions can also be easily predicted using the analytical method. Finally, a Fermi-Accelerator being excited at both ends will be discussed in detail for application. Different types of motions will be thoroughly studied for such a vibro-impact system under both same and different excitations

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit

Intrinsic Energy Localization through Discrete Gap Breathers in One-Dimensional Diatomic Granular Crystals

We present a systematic study of the existence and stability of discrete breathers that are spatially localized in the bulk of a one-dimensional chain of compressed elastic beads that interact via Hertzian contact. The chain is diatomic, consisting of a periodic arrangement of heavy and light spherical particles. We examine two families of discrete gap breathers: (1) an unstable discrete gap breather that is centered on a heavy particle and characterized by a symmetric spatial energy profile and (2) a potentially stable discrete gap breather that is centered on a light particle and is characterized by an asymmetric spatial energy profile. We investigate their existence, structure, and stability throughout the band gap of the linear spectrum and classify them into four regimes: a regime near the lower optical band edge of the linear spectrum, a moderately discrete regime, a strongly discrete regime that lies deep within the band gap of the linearized version of the system, and a regime near the upper acoustic band edge. We contrast discrete breathers in anharmonic FPU-type diatomic chains with those in diatomic granular crystals, which have a tensionless interaction potential between adjacent particles, and highlight in that the asymmetric nature of the latter interaction potential may lead to a form of hybrid bulk-surface localized solutions

Discrete Breathers in a Nonlinear Polarizability Model of Ferroelectrics

We present a family of discrete breathers, which exists in a nonlinear polarizability model of ferroelectric materials. The core-shell model is set up in its non-dimensionalized Hamiltonian form and its linear spectrum is examined. Subsequently, seeking localized solutions in the gap of the linear spectrum, we establish that numerically exact and potentially stable discrete breathers exist for a wide range of frequencies therein. In addition, we present nonlinear normal mode, extended spatial profile solutions from which the breathers bifurcate, as well as other associated phenomena such as the formation of phantom breathers within the model. The full bifurcation picture of the emergence and disappearance of the breathers is complemented by direct numerical simulations of their dynamical instability, when the latter arises.Comment: 9 pages, 7 figures, 1 tabl

Fundamentals and applications of spatial dissipative solitons in photonic devices : [Chapter 6]

We review the properties of optical spatial dissipative solitons (SDS). These are stable, self‐localized optical excitations sitting on a uniform, or quasi‐uniform, background in a dissipative environment like a nonlinear optical cavity. Indeed, in optics they are often termed “cavity solitons.” We discuss their dynamics and interactions in both ideal and imperfect systems, making comparison with experiments. SDS in lasers offer important advantages for applications. We review candidate schemes and the tremendous recent progress in semiconductor‐based cavity soliton lasers. We examine SDS in periodic structures, and we show how SDS can be quantitatively related to the locking of fronts. We conclude with an assessment of potential applications of SDS in photonics, arguing that best use of their particular features is made by exploiting their mobility, for example in all‐optical delay lines