Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can one say what should happen?

We consider a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $\Sigma$. Within the class of Filippov solutions, if $\Sigma$ is attractive, one should expect solution trajectories to slide on $\Sigma$. It is well known, however, that the classical Filippov convexification methodology is ambiguous on $\Sigma$. The situation is further complicated by the possibility that, regardless of how sliding on $\Sigma$ is taking place, during sliding motion a trajectory encounters so-called generic first order exit points, where $\Sigma$ ceases to be attractive. In this work, we attempt to understand what behavior one should expect of a solution trajectory near $\Sigma$ when $\Sigma$ is attractive, what to expect when $\Sigma$ ceases to be attractive (at least, at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature. Through analysis and experiments we will confirm some known facts, and provide some important insight: (i) when $\Sigma$ is attractive, a solution trajectory indeed does remain near $\Sigma$, viz. sliding on $\Sigma$ is an appropriate idealization (of course, in general, one cannot predict which sliding vector field should be selected); (ii) when $\Sigma$ loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of $\Sigma$; (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near $\Sigma$ as long as $\Sigma$ is attractive, and so that it will be leaving (a neighborhood of) $\Sigma$ when $\Sigma$ looses attractivity. We reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near $\Sigma$ (or sliding motion on $\Sigma$) should have been taking place.Comment: 19 figure

On Filippov solutions of discontinuous DAEs of index 1

We study discontinuous differential-algebraic equations (DDAEs) with a co-dimension 1 discontinuity manifold Σ. Our main objectives are to give sufficient conditions that allow to extend the DAE along Σ and, when this is possible, to define sliding motion (the sliding DAE) on Σ, extending Filippov construction to this DAE case. Our approach is to consider discontinuous ODEs associated to the DDAE and apply Filippov theory to the discontinuous ODEs, defining sliding/crossing solutions of the DDAE to be those inherited by the sliding/crossing solutions of the associated discontinuous ODEs. We will see that, in general, the sliding DAE on Σ is not defined unambiguously. When possible, we will consider in greater details two different methods based on Filippov's methodology to arrive at the sliding DAE. We will call these the direct approach and the Singular Perturbation Approach and we will explore advantages and disadvantages of each of them. We illustrate our development with numerical examples

Exponential dichotomy on the real line: SVD and QR methods

In this work we show when and how techniques based on the singular value decomposition (SVD) and the QR decomposition of a fundamental matrix solution can be used to infer if a system enjoys—or not—exponential dichotomy on the whole real line

Market mood, adaptive beliefs and asset price dynamics

Empirical evidence has suggested that, facing different trading strategies and complicated decision, the proportions of agents relying on particular strategies may stay at constant level or vary over time. This paper presents a simple "dynamic market fraction" model of two groups of traders, fundamentalists and trend followers, under a market maker scenario. Market mood and evolutionary adaption are characterized by fixed and adaptive switching fraction among two groups, respectively. Using local stability and bifurcation analysis, as well as numerical simulation, the role played by the key parameters in the market behaviour is examined. Particular attention is paid to the impact of the market fraction, determined by the fixed proportions of confident fundamentalists and trend followers, and by the proportion of adaptively rational agents, who adopt different strategies over time depending on realized profits. © 2005 Elsevier Ltd. All rights reserved

The Emergence ofBull and BearDynamics in a Nonlinear Model of Interacting Markets

We develop a three-dimensional nonlinear dynamic model in which the stock markets of two countries are linked through the foreign exchange market. Connections are due to the trading activity of heterogeneous speculators. Using analytical and numerical tools, we seek to explore how the coupling of the markets may affect the emergence ofbull and bearmarket dynamics. The dimension of the model can be reduced by restricting investors' trading activity, which enables the dynamic analysis to be performed stepwise, from low-dimensional cases up to the full three-dimensional model. In our paper we focus mainly on the dynamics of the one- and two- dimensional cases, with numerical experiments and some analytical results, and also show that the main features persist in the three-dimensional model

A novel RNA polymerase III transcription factor fraction that is not required for template commitment.

Abstract We have identified and partially characterized a novel class III transcription factor fraction (TFIIIE) from yeast nuclear extracts. TFIIIE is functionally distinct from the standard yeast transcription factor fractions, TFIIIB and TFIIIC. It is also different from either of the TFIIIB subfractions, B' and B". TFIIIE is essential for specific transcription of both tRNA and 5 S RNA genes, its activity is sensitive to proteinase K, and it exhibits an apparent sedimentation coefficient of 4.0 S when analyzed on glycerol gradients. In the case of a tRNA gene, TFIIIE does not play a role in the formation of stable preinitiation complexes containing TFIIIB and TFIIIC. It is required for single as well as multiple rounds of transcription, however. Thus, TFIIIE is involved in the utilization of stable transcription complexes, but its action is not restricted to reinitiation events

A bit stickier, a bit slower, a lot stiffer: Specific vs. nonspecific binding of gal4 to dna

Transcription factors regulate gene activity by binding specific regions of genomic DNA thanks to a subtle interplay of specific and nonspecific interactions that is challenging to quantify. Here, we exploit Reflective Phantom Interface (RPI), a label-free biosensor based on optical reflectivity, to investigate the binding of the N-terminal domain of Gal4, a well-known gene regulator, to double-stranded DNA fragments containing or not its consensus sequence. The analysis of RPI-binding curves provides interaction strength and kinetics and their dependence on temperature and ionic strength. We found that the binding of Gal4 to its cognate site is stronger, as expected, but also markedly slower. We performed a combined analysis of specific and nonspecific binding— equilibrium and kinetics—by means of a simple model based on nested potential wells and found that the free energy gap between specific and nonspecific binding is of the order of one kcal/mol only. We investigated the origin of such a small value by performing all-atom molecular dynamics simulations of Gal4–DNA interactions. We found a strong enthalpy–entropy compensation, by which the binding of Gal4 to its cognate sequence entails a DNA bending and a striking conformational freezing, which could be instrumental in the biological function of Gal4

Introduction

This collected volume gives a concise account of the most relevant scientific results of the COST Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation", a four-year project supported by COST (European Cooperation in Science and Technology). It is divided into three parts reflecting the different perspectives under which complex spatial economic systems have been studied: (i) the Macro perspective looks at the interactions among international or regional trading partners; (ii) the Meso perspective considers the functioning of (financial, labour) markets as social network structures; and, finally, (iii) the Micro perspective focuses on the strategic choices of single firms and households. This Volume points also at open issues to be addressed in future research