A horseshoe in the dynamics of a forced beam

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Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups

Abstract not availabl

Nonlinear Propagation of Light in One Dimensional Periodic Structures

We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction which varies periodically along its length. The wavelength of light is selected to be in resonance with the periodic structure (Bragg resonance). The AMLE system considered incorporates the effects non-instantaneous response of the medium to the electromagnetic field (chromatic or material dispersion), the periodic structure (photonic band dispersion) and nonlinearity. We present a detailed discussion of the role of these effects individually and in concert. We derive the nonlinear coupled mode equations (NLCME) which govern the envelope of the coupled backward and forward components of the electromagnetic field. We prove the validity of the NLCME description and give explicit estimates for the deviation of the approximation given by NLCME from the {\it exact} dynamics, governed by AMLE. NLCME is known to have gap soliton states. A consequence of our results is the existence of very long-lived {\it gap soliton} states of AMLE. We present numerical simulations which validate as well as illustrate the limits of the theory. Finally, we verify that the assumptions of our model apply to the parameter regimes explored in recent physical experiments in which gap solitons were observed.Comment: To appear in The Journal of Nonlinear Science; 55 pages, 13 figure

Agglomeration externalities and 1981-2006 regional growth in Brazil

This paper focuses on manufacturing employment growth across the 26 states of Brazil. We employ the Glaeser et al. (1992) approach to identify the role played by knowledge externalities in growth and convergence. To assess robustness of the results, we compare cross-section models, dynamic panel models and pooled-periods fixed-effect models. We find that cross-section models confirm the positive impact of Porter’s and Jacobs’ competition externalities on growth, whereas dynamic panel models and pooled-periods fixed-effect models are consistent with the predictions of Marshall-Arrow-Romer and Porter regarding the role of specialisation in manufacturing vis-à-vis other employment. The results provide new insights into the rapid growth since 1981 in particularly the North and Centre West of Brazil

A century of the evolution of the urban system in Brazil

In this paper, we study the hitherto unexplored evolution of the size distribution of 185 urban areas in Brazil between 1907 and 2008. We find that the power law parameter of the size distribution of the 100 largest urban areas increases from 0.63 in 1907 to 0.89 in 2008, which confirms an agglomeration process in which the size distribution has become more unequal. A panel fixed effects model pooling the same range of urban size distributions provides a power law parameter equal to 0.53, smaller than those from cross-sectional estimation. Clearly, Zipf’s Law is rejected. The lognormal distribution fits the city size distribution quite well until the 1940s, but since then applies to small and medium size cities only. These results are consistent with our understanding of historical-political and socio-economic processes that have shaped the development of Brazilian cities

Melnikov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems

We start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates. We use Kolmogorov–Arnold–Moser (KAM) theory to show that some of the resulting tori persist under small perturbations and use a vector of Melnikov integrals to show that, under suitable hypotheses, their stable and unstable manifolds intersect transversely. This transverse intersection is ultimately responsible for Arnold diffusion on each energy surface. The method is applied to a pendulum–oscillator system

A model of meta-population dynamics for North Sea and West of Scotland cod - the dynamic consequences of natal fidelity

It is clear from a variety of data that cod (Gadus morhua) in the North Sea do not constitute a homogeneous population that will rapidly redistribute in response to local variability in exploitation. Hence, local exploitation has the potential to deplete local populations, perhaps to the extent that depensation occurs and recovery is impossible without recolonisation from other areas, with consequent loss of genetic diversity. The oceanographic, biological and behavioural processes which maintain the spatial population structures are only partly understood, and one of the key unknown factors is the extent to which codexhibit homing migrations to natal spawning areas. Here, we describe a model comprising 10 interlinked demes of cod in European waters, each representing groups of fish with a common natal origin. The spawning locations of fish in each deme are governed by a variety of rules concerning oceanographic dispersal, migration behaviour and straying. We describe numerical experiments with the model and comparisons with observations, which lead us to conclude that active homing is probably not necessary to explain some of the population structures of European cod. Separation of some sub-populations is possible through distance and oceanographic processes affecting the dispersal of eggs and larvae. However, other evidence suggests that homing may be a necessary behaviour to explain the structure of other sub-populations. Theconsequences for fisheries management of taking into account spatial population structuring are complicated. For example, recovery or recolonisation strategies require consideration not only of mortality rates in the target area for restoration, but also in the source areas for the recruits which may be far removed depending on the oceanography. The model has an inbuilt capability to address issues concerning the effects of climate change, including temperature change, on spatial patterns of recruitment, development and population structure in cod

CHOOSING TO ADVANCE: MOVING TEACHERS IN THE CHAPEL HILL CARRBORO CITY SCHOOLS TO A KNOWLEDGE AND SKILLS-BASED COMPENSATION SYSTEM

This study examines the design decisions, or choice architecture, that was made in planning and implementing a knowledge and skills-based career ladder in the Chapel Hill Carrboro City Schools (CHCCS). At the start of the 2014-2015 school year leadership of the CHCCS set about designing a knowledge and skills-based career ladder with the goals of reimagining professional development in the district and creating a system that allowed for teacher career and financial advancement. This system would become known as Project ADVANCE. At the end of the first year of design work a basic outline of this program had been created and shared with staff. Survey results showed that the staff of the CHCCS was mostly anxious about the implementation of Project ADVANCE (Pepper, 2015). This Project ADVANCE Implementation Team was formed in July of 2015 and this 13-member team was tasked with designing the specifics of the program and responding to the concerns expressed by staff in the spring of 2015. This study analyzes the design decisions, or choice architecture, made by the Implementation Team and how those decisions were able to improve perception of Project ADVANCE such that when staff members were given the option to opt-in or out of the first year of implementation, over 75% of eligible staff chose to opt-in. These design decisions are specifically analyze through the lenses of Thaler and Sunstein's 2009 book about choice architecture, Nudge, Langley, Moen, Nolan, Nolan, Norman, and Provost's 2009 The Improvement Guide, and Pink's 2009 work on motivation, Drive. By examining the design decisions made through these three lenses one can draw conclusions on how these decisions helped to improve perception of Project ADVANCE among CHCCS staff. In addition recommendations can be drawn from this change effort that can be related to other school districts or major institutions that are looking at making large, wide-ranging, and potentially controversial change

Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

Dynamical Systems and Invariant Manifolds

We review some basic terminology in dynamical systems with the purpose of bridging some of the communication gaps that may exist between mathematicians and engineers at this conference. Recent results on panel flutter and on the existence of horseshoes in the dynamics of a forced beam are briefly sketched to illustrate some of the concepts of interest to both groups