An adaptive Newton-method based on a dynamical systems approach

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.

A Dynamical Tool to Study the Cultural Context of Conflict Escalation

The present article describes research in progress which is developing a simple, replicable methodology aimed at identifying the regularities and specificity of human behavior in conflict escalation and de-escalation prooesses. These research efforts will ultimately be used to study conflict dynamics across cultures. The experimental data collected through this methodology, together with case studies and aggregated, time-series macro data are key for identifying relevant parameters, systems' properties, and micromechanisms defining the behavior of naturally occurring conflict escalation and de-escalation dynamics. This, in turn, is critical for the development of realistic, empirically supported computational models. The article outlines the theoretical assumptions of Dynamical Systems Theory with regard to conflict dynamics, with an emphasis on the process of conflict escalation and de-escalation. Next, work on a methodology for empirical study of escalation processes from a DST perspective is outlined. Specifically, the development of a progressive scenario methodology designed to map escalation sequences, together with anexample of a preliminary study based on the proposed researcb paradigm, is presented. Implications of the approach for the study of culture are discussed.

Uncovering the dynamics of cardiac systems using stochastic pacing and frequency domain analyses

Alternans of cardiac action potential duration (APD) is a well-known arrhythmogenic mechanism which results from dynamical instabilities. The propensity to alternans is classically investigated by examining APD restitution and by deriving APD restitution slopes as predictive markers. However, experiments have shown that such markers are not always accurate for the prediction of alternans. Using a mathematical ventricular cell model known to exhibit unstable dynamics of both membrane potential and Ca2+ cycling, we demonstrate that an accurate marker can be obtained by pacing at cycle lengths (CLs) varying randomly around a basic CL (BCL) and by evaluating the transfer function between the time series of CLs and APDs using an autoregressive-moving-average (ARMA) model. The first pole of this transfer function corresponds to the eigenvalue (λalt) of the dominant eigenmode of the cardiac system, which predicts that alternans occurs when λalt≤−1. For different BCLs, control values of λalt were obtained using eigenmode analysis and compared to the first pole of the transfer function estimated using ARMA model fitting in simulations of random pacing protocols. In all versions of the cell model, this pole provided an accurate estimation of λalt. Furthermore, during slow ramp decreases of BCL or simulated drug application, this approach predicted the onset of alternans by extrapolating the time course of the estimated λalt. In conclusion, stochastic pacing and ARMA model identification represents a novel approach to predict alternans without making any assumptions about its ionic mechanisms. It should therefore be applicable experimentally for any type of myocardial cell.