Identification Of Bifurcation Points Research Articles

Digital Bifurcation Analysis of Internet Congestion Control Protocols

Digital bifurcation analysis is a new algorithmic method for exploring how the behavior of a parameter-dependent discrete system varies with a change in its parameters and, in particular, for the identification of bifurcation points where such variation becomes dramatic. We have developed the method in an analogy with the traditional bifurcation theory and have successfully applied it to models taken from systems biology. In this paper, we report on the application of the digital bifurcation analysis for analyzing the stability of internet congestion control protocols by inspecting their attractor bifurcations. In contrast to the analytical methods, our approach allows fully automated analysis. We compared the robustness of the basic Random Early Drop (RED) approach with four substantially different extensions, namely gentle, adaptive, gradient descent, and integral feedback RED. The basic RED protocol is well known to exhibit unstable behavior when parameters are varied. In the case of adaptive and gradient descent RED protocol, the analysis showed significant improvements in stability, whereas in the results for gentle and integral feedback RED protocols the improvement was negligible. We performed a series of model simulations, the results of which were in accordance with our bifurcation analysis. Based on our results, we can recommend both adaptive and gradient descent RED to improve the robustness of the RED protocol.

An eigenanalysis-based bifurcation indicator proposed in the framework of a reduced-order modeling technique for non-linear structural analysis

The Koiter–Newton method is a reduced order modeling technique which allows us to trace efficiently the entire equilibrium path of a non-linear structural analysis. In the framework of buckling the method is capable to handle snap-back and snap-through phenomena but may fail to predict reliably bifurcation branches along the equilibrium path. In this contribution we extend the original Koiter–Newton approach with a reliable and accurate bifurcation indicator which is based on an eigenanalysis of the reduced order tangent stiffness matrix. The proposed indicator has a negligible numerical effort since all computations refer to the reduced order model which is typically of very small dimension. The extension allows the identification of bifurcation points and a tracing of corresponding bifurcation branches in each sector of the equilibrium path. The performance of the method in terms of reliability, accuracy and computational effort is demonstrated with several examples.

Identification of Bifurcation Points Occurring on the Mechanism Paths of Deployable Structures

This paper is concerned with the identification of bifurcation points occurring on the mechanism paths of a linkage, in which the structural system consists of links and revolute joints. The way that the links are assembled is mathematically described by compatibility conditions. The present paper mainly deals with the kinematic paths of mechanism that goes through special configurations of kinematic bifurcations. In general, such special configurations are called uncertainty configurations or singular configurations. The Jacobian derived from the compatibility conditions has been used as a measure of the distance away from the singular configurations. The Jacobian does not always give sufficient results for the detection of bifurcation points. This suggests that it is necessary to consider higher-order infinitesimals for examining approximate solutions of compatibility conditions. The Hessian matrix derived from the Jacobi matrix is considered to be one of the promising measure and may clarify the nature of the singular configuration as to whether the system is at a bifurcation point or not. In this paper, the theory behind the kinematic bifurcation of mechanisms of a linkage is explained and the numerical analysis of illustrative examples is discussed in detail.

The use of boundary locus plots in the identification of bifurcation points in numerical approximation of delay differential equations

We are interested in nonlinear delay differential equations which have a Hopf bifurcation. We assume zero is a steady state for the problem, and so a Hopf bifurcation point lies on the boundary of the region of asymptotic stability for the zero solution. We investigate whether discrete versions of the nonlinear delay differential equation also exhibit Hopf bifurcations. We use the boundary locus method as a tool both for the delay differential equation and for numerical analogues. We use the information obtained about the stability domain to assist in identifying Hopf bifurcations. We demonstrate the following: • For certain linear multistep methods, the boundary of the region of stability for the zero solution of the original equation is approximated by the boundary of the region of stability for the zero solution of the numerical analogue equation to the order of the method. • The boundary locus method enables us to determine precise parameter values at which any Hopf bifurcations arise in the discrete equations. We prove that Hopf bifurcation points for the true equation are approximated to the order of the method by corresponding points in the discrete scheme. • Further calculations are necessary to determine the precise nature of bifurcation points identified in this way.

Detecting and Traversing Bifurcation Points in Nonlinear Structural Analysis

We investigate procedures for detecting and traversing bifurcation points that occur in the geometrically nonlinear analysis of elastic structures. In particular, we study a family of test functions for sharp detection and identification of bifurcation points. To accomplish branch switching without recourse to artifical imperfections, the selection of appropriate predictors and correctors which exclude the known branch from the converged solution is considered. These techniques are tried and assesed on several three-dimensional truss structures that exhibit bifurcation behavior.

Nonlinear Dynamic Modelling of Spatial Interactions

Attention is focused on nonlinear modelling in space and time, with particular emphasis on migration behaviour. A pedagogic overview is provided of the use of dynamic models for migration behaviour based on synergetic dynamic effects. This is followed by an attempt at identifying bifurcation points in the framework of a simple residential choice model. The paper begins with a discussion of various issues related to dynamic (spatial) economic models. Then the master equation approach (and the related mean value model) is discussed as an interesting and promising dynamic modelling framework for spatial interactions with synergetic effects. Next, to illustrate the potential of this approach, a framework of a nonlinear dynamic migration model is presented. Last, some results of a simulation experiment with a simple discrete choice model are also presented, in which particular emphasis is placed on the identification of bifurcation points.

Examination of an Exponential Model of Conduction Through the Human Atrioventricular Node *

The atrioventricular node (AVN) has been modeled by relating output (A2H2 or H1H2) to input (A1A2) where A and H are atrial and His bundle electrograms during fixed rate atrial pacing (A1A1) or with an extrastimulus (A2). (Formula: see text) This study examined this model in 61 nonselected patients, specifically for AVN (in)stability and the possibility of multiple pathways. After programmed atrial stimulation at two basic cycle lengths of 600 ms and 462 ms, A1H1, A2H2 and H1H2 were digitized and plotted as a function of A1A2. Seven of 104 trials were rejected as SD. A1H1 was greater than 15 ms, suggesting AVN instability. Another 26 and 34 plots, respectively, of A2H2 and H1H2 were rejected because of inadequate data. In the remainder, goodness of fit of the single exponentials was tested statistically in three ways: R2, the runs test, and the Kendall rank coefficient test. Results were compared with an electrophysiologist who examined plots for one or more pathways (either discontinuous curves or slope change in a continuous curve). Single exponentials were successfully fitted (by runs test) in 44/71 and 34/63 of A2H2 and H1H2 plots, respectively, usually in accordance with the cardiologist. Discordance between computations and the cardiologist could be attributed to data scatter and lack of a sufficiently rigid stimulation protocol. The identification of bifurcation points in the presence of multiple pathways, particularly when manifest as a change in slope (approximately 6% of trials) rather than discontinuity of plots (approximately 20% of trials) remains an outstanding problem.