Mode-locking in advection-reaction-diffusion systems: an invariant manifold perspective

Fronts propagating in two-dimensional advection-reaction-diffusion (ARD) systems exhibit rich topological structure. When the underlying fluid flow is periodic in space and time, the reaction front can lock to the driving frequency. We explain this mode-locking phenomenon using so-called burning invariant manifolds (BIMs). In fact, the mode-locked profile is delineated by a BIM attached to a relative periodic orbit (RPO) of the front element dynamics. Changes in the type (and loss) of mode-locking can be understood in terms of local and global bifurcations of the RPOs and their BIMs. We illustrate these concepts numerically using a chain of alternating vortices in a channel geometry.Comment: 9 pages, 13 figure

Lessons From India in Organizational Innovation: A Tale of Two Heart Hospitals

Recent discussions in health reform circles have pinned great hopes on the prospect of innovation as the solution to the high-cost, inadequate-quality U.S. health system. But U.S. health care institutions--insurers, providers and specialists--have ceded leadership in innovation to Indian hospitals such as Care Hospital in Hyderabad and the Fortis Hospitals around New Delhi, which have U.S.-trained doctors and can perform open heart surgery for $6000 (compared to$100,000 in the United States). The Indian success is a window into America\u27s stalemate with inflating costs and stagnant innovation

Topological Chaos in a Three-Dimensional Spherical Fluid Vortex

In chaotic deterministic systems, seemingly stochastic behavior is generated by relatively simple, though hidden, organizing rules and structures. Prominent among the tools used to characterize this complexity in 1D and 2D systems are techniques which exploit the topology of dynamically invariant structures. However, the path to extending many such topological techniques to three dimensions is filled with roadblocks that prevent their application to a wider variety of physical systems. Here, we overcome these roadblocks and successfully analyze a realistic model of 3D fluid advection, by extending the homotopic lobe dynamics (HLD) technique, previously developed for 2D area-preserving dynamics, to 3D volume-preserving dynamics. We start with numerically-generated finite-time chaotic-scattering data for particles entrained in a spherical fluid vortex, and use this data to build a symbolic representation of the dynamics. We then use this symbolic representation to explain and predict the self-similar fractal structure of the scattering data, to compute bounds on the topological entropy, a fundamental measure of mixing, and to discover two different mixing mechanisms, which stretch 2D material surfaces and 1D material curves in distinct ways.Comment: 14 pages, 11 figure

Frozen reaction fronts in steady flows: a burning-invariant-manifold perspective

The dynamics of fronts, such as chemical reaction fronts, propagating in two-dimensional fluid flows can be remarkably rich and varied. For time-invariant flows, the front dynamics may simplify, settling in to a steady state in which the reacted domain is static, and the front appears "frozen". Our central result is that these frozen fronts in the two-dimensional fluid are composed of segments of burning invariant manifolds---invariant manifolds of front-element dynamics in $xy\theta$-space, where $\theta$ is the front orientation. Burning invariant manifolds (BIMs) have been identified previously as important local barriers to front propagation in fluid flows. The relevance of BIMs for frozen fronts rests in their ability, under appropriate conditions, to form global barriers, separating reacted domains from nonreacted domains for all time. The second main result of this paper is an understanding of bifurcations that lead from a nonfrozen state to a frozen state, as well as bifurcations that change the topological structure of the frozen front. Though the primary results of this study apply to general fluid flows, our analysis focuses on a chain of vortices in a channel flow with an imposed wind. For this system, we present both experimental and numerical studies that support the theoretical analysis developed here.Comment: 21 pages, 30 figure

From Gatekeeping to Engagement: A Multicontextual, Mixed Method Study of Student Academic Engagement in Introductory STEM Courses.

The lack of academic engagement in introductory science courses is considered by some to be a primary reason why students switch out of science majors. This study employed a sequential, explanatory mixed methods approach to provide a richer understanding of the relationship between student engagement and introductory science instruction. Quantitative survey data were drawn from 2,873 students within 73 introductory science, technology, engineering, and mathematics (STEM) courses across 15 colleges and universities, and qualitative data were collected from 41 student focus groups at eight of these institutions. The findings indicate that students tended to be more engaged in courses where the instructor consistently signaled an openness to student questions and recognizes her/his role in helping students succeed. Likewise, students who reported feeling comfortable asking questions in class, seeking out tutoring, attending supplemental instruction sessions, and collaborating with other students in the course were also more likely to be engaged. Instructional implications for improving students' levels of academic engagement are discussed

Invariant manifolds and the geometry of front propagation in fluid flows

Recent theoretical and experimental work has demonstrated the existence of one-sided, invariant barriers to the propagation of reaction-diffusion fronts in quasi-two-dimensional periodically-driven fluid flows. These barriers were called burning invariant manifolds (BIMs). We provide a detailed theoretical analysis of BIMs, providing criteria for their existence, a classification of their stability, a formalization of their barrier property, and mechanisms by which the barriers can be circumvented. This analysis assumes the sharp front limit and negligible feedback of the front on the fluid velocity. A low-dimensional dynamical systems analysis provides the core of our results.Comment: 14 pages, 11 figures. To appear in Chaos Focus Issue: Chemo-Hydrodynamic Patterns and Instabilities (2012