Symmetry breaking in dynamical systems

Symmetry breaking bifurcations and dynamical systems have obtained a lot of attention over the last years. This has several reasons: real world applications give rise to systems with symmetry, steady state solutions and periodic orbits may have interesting patterns, symmetry changes the notion of structural stability and introduces degeneracies into the systems as well as geometric simplifications. Therefore symmetric systems are attractive to those who study specific applications as well as to those who are interested in a the abstract theory of dynamical systems. Dynamical systems fall into two classes, those with continuous time and those with discrete time. In this paper we study only the continuous case, although the discrete case is as interesting as the continuous one. Many global results were obtained for the discrete case. Our emphasis are heteroclinic cycles and some mechanisms to create them. We do not pursue the question of stability. Of course many studies have been made to give conditions which imply the existence and stability of such cycles. In contrast to systems without symmetry heteroclinic cycles can be structurally stable in the symmetric case. Sometimes the various solutions on the cycle get mapped onto each other by group elements. Then this cycle will reduce to a homoclinic orbit if we project the equation onto the orbit space. Therefore techniques to study homoclinic bifurcations become available. In recent years some efforts have been made to understand the behaviour of dynamical systems near points where the symmetry of the system was perturbed by outside influences. This can lead to very complicated dynamical behaviour, as was pointed out by several authors. We will discuss some of the technical difficulties which arise in these problems. Then we will review some recent results on a geometric approach to this problem near steady state bifurcation points

Existence and Stability of N-Pulses on Optical Fibers with Phase-Sensitive Amplifiers

The propagation of pulses in optical communication systems in which attenuation is compensated by phase-sensitive amplifiers is investigated. A central issue is whether optical fibers are capable of carrying several pieces of information at the same time. In this paper, multiple pulses are shown to exist for a fourth-order nonlinear diffusion model due to Kutz and co-workers [10]. Moreover, criteria are derived for determining which of these pulses are stable. The pulses arise in a reversible orbit-flip, a homoclinic bifurcation investigated here for the first time. Numerical simulations are used to study multiple pulses far away from the actual bifurcation point. They confirm that properties of the multiple pulses including their stability are surprisingly well predicted by the analysis carried out near the bifurcation

1991 Summer Study Program in Geophysical Fluid Dynamics : patterns in fluid flow

The GFD program in 1991 focused on pattern forming processes in physics and geophysics. The pricipallecturer, Stephan Fauve, discussed a variety of systems, including our old favorite, Rayleigh-Bénard convection, but passing on to exotic examples such as vertically vibrated granular layers. Fauve's lectures emphasize a unified theoretical viewpoint based on symmetry arguments. Patterns produced by instabilties can be described by amplitude equations, whose form can be deduced by symmetry arguments, rather than the asymptotic expansions that have been the staple of past Summer GFD Programs. The amplitude equations are far simpler than the complete equations of motion, and symetry arguments are easier than asymptotic expansions. Symmetry arguments also explain why diverse systems are often described by the same amplitude equation. Even for granular layers, where there is not a universaly accepted continuum description, the appropnate amplitude equation can often be found using symmetry arguments and then compared with experiment. Our second speaker, Daniel Rothan, surveyed the state of the art in lattice gas computations. His lectures illustrate the great utility of these methods in simulating the flow of complex multiphase fluids, particularly at low Reynolds numbers. The lattice gas simulations reveal a complicated phenomenology much of which awaits analytic exploration. The fellowship lectures cover broad ground and reflect the interests of the staff members associated with the program. They range from the formation of sand dunes, though the theory of lattice gases, and on to two dimensional-turbulence and convection on planetary scales. Readers desiring to quote from these report should seek the permission of the authors (a partial list of electronic mail addresses is included on page v). As in previous years, these reports are extensively reworked for publication or appear as chapters in doctoral theses. The task of assembling the volume in 1991 was at first faciltated by our newly acquired computers, only to be complicated by hurricane Bob which severed electric power to Walsh Cottage in the final hectic days of the Summer.Funding was provided by the National Science Foundation through Grant No. OCE 8901012

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute

Air-data estimation for air-breathing hypersonic vehicles

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1996.Includes bibliographical references (p. 194-198).by Bryan Heejin Kang.Ph.D

Nonlinear optics

Nonlinear light-matter interactions have been drawing attention of physicists since the 1960's. Quantum mechanics played a significant role in their description and helped to derive important formulas showing the dependence on the intensity of the electromagnetic field. High intensity light is able to generate second and third harmonics which translates to generation of electromagnetic field with multiples of the original frequency. In comparison with the linear behaviour of light, the nonlinear interactions are smaller in scale. This makes perturbation methods well suited for obtaining solutions to equations in nonlinear optics. In particular, the method of multiple scales is deployed in paper 3, where it is used to solve nonlinear dispersive wave equations. The key difference in our multiple scale solution is the linearity of the amplitude equation and a complex valued frequency of the mode. Despite the potential ill-posedness of the amplitude equation, the multiple scale solution remained a valid approximation of the solution to the original model. The results showed great potential of this method and its promising wider applications. Other methods use pseudo-spectral methods which require an orthogonal set of eigenfunctions (modes) used to create a substitute for the usual Fourier transform. This mode transform is only useful if it succeeds to represent target functions well. Papers 1 and 2 deal with investigating such modes called resonant and leaky modes and their ability to construct a mode transform. The modes in the first paper are the eigenvalues for a quantum mechanical system where an external radiation field is used to excite an electron trapped in an electrical potential. The findings show that the resonant mode expansion converges inside the potential independently of its depth. Equivalently, leaky modes are obtained in paper 2 which are in close relation to resonant modes. Here, the modes emerge from a system where a channel is introduced with transparent boundaries for simulation of one-directional optical beam propagation. Artificial index material is introduced outside the channel which gives rise to leaky modes associated with such artificial structure. The study is showing that leaky modes are well suited for function representation and thus solving the nonlinear version of this problem. In addition, the transparent boundary method turns out to be useful for spectral propagators such as the unidirectional pulse propagation equation in contrast to a perfectly matched layer