Scholars@Duke
David G. Schaeffer

David G. Schaeffer

James B. Duke Distinguished Professor Emeritus of Mathematics
Mathematics
Box 90320, Durham, NC 27708-0320
231 Physics Bldg, Durham, NC 27708

Selected Publications

The hanging thin rod: a singularly perturbed eigenvalue problem

Journal Article SIAM Sppl. Math. · July 2010 Cite

Bifurcations in a modulation equation for alternans in a cardiac fiber

Journal Article ESAIM: Mathematical Modelling and Numerical Analysis · January 1, 2010 While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either station ... Full text Open Access Cite

Spectrum of a linearized amplitude equation for alternans in a cardiac fiber

Journal Article SIAM Journal on Applied Mathematics · December 1, 2008 Under rapid periodic pacing, cardiac cells typically undergo a period-doubling bifurcation in which action potentials of short and long duration alternate with one another. If these action potentials propagate in a fiber, the short-long alternation may suf ... Full text Open Access Cite

Steady and intermittent slipping in a model of landslide motion regulated by pore-pressure feedback

Journal Article SIAM Journal on Applied Mathematics · December 1, 2008 This paper studies a parsimonious model of landslide motion, which consists of the one-dimensional diffusion equation (for pore pressure) coupled through a boundary condition to a first-order ODE (Newton's second law). Velocity weakening of sliding frictio ... Full text Cite

Electrical waves in a one-dimensional model of cardiac tissue

Journal Article SIAM Applied Dynamical Systems · December 2008 Full text Cite

Shortening of action potential duraction near an insulating boundary

Journal Article Math Medicine and Biology · 2008 Cite

Asymptotic approximation of an ionic model for cardiac restitution.

Journal Article Nonlinear dynamics · January 2008 Cardiac restitution has been described both in terms of ionic models-systems of ODE's-and in terms of mapping models. While the former provide a more fundamental description, the latter are more flexible in trying to fit experimental data. Recently we prop ... Full text Cite

Alternate Pacing of Border-Collision Period-Doubling Bifurcations.

Journal Article Nonlinear dynamics · November 2007 Unlike classical bifurcations, border-collision bifurcations occur when, for example, a fixed point of a continuous, piecewise C1 map crosses a boundary in state space. Although classical bifurcations have been much studied, border-collision bifurcations a ... Full text Cite

Period-doubling bifurcation to alternans in paced cardiac tissue: crossover from smooth to border-collision characteristics.

Journal Article Physical review letters · August 2007 We investigate, both experimentally and theoretically, the period-doubling bifurcation to alternans in heart tissue. Previously, this phenomenon has been modeled with either smooth or border-collision dynamics. Using a modification of existing experimental ... Full text Cite

Small-Signal Amplification of Period-Doubling Bifurcations in Smooth Iterated Maps.

Journal Article Nonlinear dynamics · June 2007 Various authors have shown that, near the onset of a period-doubling bifurcation, small perturbations in the control parameter may result in much larger disturbances in the response of the dynamical system. Such amplification of small signals can be measur ... Full text Cite

An ionically based mapping model with memory for cardiac restitution.

Journal Article Bulletin of mathematical biology · February 2007 Many features of the sequence of action potentials produced by repeated stimulation of a patch of cardiac muscle can be modeled by a 1D mapping, but not the full behavior included in the restitution portrait. Specifically, recent experiments have found tha ... Full text Cite

Force distribution in granular media

Journal Article PRE · 2005 Cite

Secondary circulation in granular flow through nonaxisymmetric hoppers

Journal Article SIAM Journal on Applied Mathematics · June 7, 2004 Jenike's radial solution, widely used in the design of materials-handling equipment, is a similarity solution of steady-state continuum equations for the flow under gravity of granular material through an infinite, right-circular cone. In this paper we stu ... Full text Cite

A two-current model for the dynamics of cardiac membrane.

Journal Article Bulletin of mathematical biology · September 2003 In this paper we introduce and study a model for electrical activity of cardiac membrane which incorporates only an inward and an outward current. This model is useful for three reasons: (1) Its simplicity, comparable to the FitzHugh-Nagumo model, makes it ... Full text Cite

Condition for alternans and stability of the 1:1 response pattern in a "memory" model of paced cardiac dynamics.

Journal Article Physical review. E, Statistical, nonlinear, and soft matter physics · March 2003 We analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration, thereby containing some degree of "memory." The ... Full text Cite

Review of W. Cheney's "Analysis for applied mathematics"

Journal Article Amer. Math Monthly · 2003 Cite

One-dimensional solutions of an elastoplasticity model of granular material

Journal Article Math. Models and Methods in Appl. Sciences · 2003 Cite

Analysis of the Fenton-Karma model through an approximation by a one-dimensional map.

Journal Article Chaos (Woodbury, N.Y.) · December 2002 The Fenton-Karma model is a simplification of complex ionic models of cardiac membrane that reproduces quantitatively many of the characteristics of heart cells; its behavior is simple enough to be understood analytically. In this paper, a map is derived t ... Full text Cite

Directed force chain networks and stress response in static granular materials.

Journal Article The European physical journal. E, Soft matter · April 2002 A theory of stress fields in two-dimensional granular materials based on directed force chain networks is presented. A general Boltzmann equation for the densities of force chains in different directions is proposed and a complete solution is obtained for ... Full text Cite

Granular Flow Past a Binsert

Journal Article Report to Jenike & Johanson, Inc. · January 7, 1997 Cite