A network dynamics approach to chemical reaction networks

A crisp survey is given of chemical reaction networks from the perspective of general nonlinear network dynamics, in particular of consensus dynamics. It is shown how by starting from the complex-balanced assumption the reaction dynamics governed by mass action kinetics can be rewritten into a form which allows for a very simple derivation of a number of key results in chemical reaction network theory, and which directly relates to the thermodynamics of the system. Central in this formulation is the definition of a balanced Laplacian matrix on the graph of chemical complexes together with a resulting fundamental inequality. This directly leads to the characterization of the set of equilibria and their stability. Both the form of the dynamics and the deduced dynamical behavior are very similar to consensus dynamics. The assumption of complex-balancedness is revisited from the point of view of Kirchhoff's Matrix Tree theorem, providing a new perspective. Finally, using the classical idea of extending the graph of chemical complexes by an extra 'zero' complex, a complete steady-state stability analysis of mass action kinetics reaction networks with constant inflows and mass action outflows is given.Comment: 18 page

Ab initio Modelling of the Early Stages of Precipitation in Al-6000 Alloys

Age hardening induced by the formation of (semi)-coherent precipitate phases is crucial for the processing and final properties of the widely used Al-6000 alloys. Early stages of precipitation are particularly important from the fundamental and technological side, but are still far from being fully understood. Here, an analysis of the energetics of nanometric precipitates of the meta-stable $\beta''$ phases is performed, identifying the bulk, elastic strain and interface energies that contribute to the stability of a nucleating cluster. Results show that needle-shape precipitates are unstable to growth even at the smallest size $\beta''$ formula unit, i.e. there is no energy barrier to growth. The small differences between different compositions points toward the need for the study of possible precipitate/matrix interface reconstruction. A classical semi-quantitative nucleation theory approach including elastic strain energy captures the trends in precipitate energy versus size and composition. This validates the use of mesoscale models to assess stability and interactions of precipitates. Studies of smaller 3d clusters also show stability relative to the solid solution state, indicating that the early stages of precipitation may be diffusion-limited. Overall, these results demonstrate the important interplay among composition-dependent bulk, interface, and elastic strain energies in determining nanoscale precipitate stability and growth

Empirical Analysis of the Necessary and Sufficient Conditions of the Echo State Property

The Echo State Network (ESN) is a specific recurrent network, which has gained popularity during the last years. The model has a recurrent network named reservoir, that is fixed during the learning process. The reservoir is used for transforming the input space in a larger space. A fundamental property that provokes an impact on the model accuracy is the Echo State Property (ESP). There are two main theoretical results related to the ESP. First, a sufficient condition for the ESP existence that involves the singular values of the reservoir matrix. Second, a necessary condition for the ESP. The ESP can be violated according to the spectral radius value of the reservoir matrix. There is a theoretical gap between these necessary and sufficient conditions. This article presents an empirical analysis of the accuracy and the projections of reservoirs that satisfy this theoretical gap. It gives some insights about the generation of the reservoir matrix. From previous works, it is already known that the optimal accuracy is obtained near to the border of stability control of the dynamics. Then, according to our empirical results, we can see that this border seems to be closer to the sufficient conditions than to the necessary conditions of the ESP.Comment: 23 pages, 14 figures, accepted paper for the IEEE IJCNN, 201

Fuzzy Approach to Conflict Analysis

Based on the concept of fuzzy sets and fuzzy relations, in this paper, a new approach is presented for modeling and analyzing conflicts. In analyzing conflicts, it is fundamental to evaluate feasible outcomes according to the preference of each player. A fuzzy preference matrix is first defined to evaluate preference relations between outcomes for each player. Several actions and reactions of players are investigated, and a new method of stability analysis is then proposed to derive the grades of membership of stability, instability and equilibrium. The new approach determines a different set of equilibria, depending on the fuzzy environment and the threshold

Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities

The Lyapunov exponent characterizes the asymptotic behavior of long matrix products. Recognizing scenarios where the Lyapunov exponent is strictly positive is a fundamental challenge that is relevant in many applications. In this work we establish a novel tool for this task by deriving a quantitative lower bound on the Lyapunov exponent in terms of a matrix sum which is efficiently computable in ergodic situations. Our approach combines two deep results from matrix analysis --- the $n$-matrix extension of the Golden-Thompson inequality and the Avalanche-Principle. We apply these bounds to the Lyapunov exponents of Schr\"odinger cocycles with certain ergodic potentials of polymer type and arbitrary correlation structure. We also derive related quantitative stability results for the Lyapunov exponent near aligned diagonal matrices and a bound for almost-commuting matrices.Comment: 46 pages; comments welcom

Modeling, analysis, and experimentation of chaos in a switched reluctance drive system

In this brief, modeling, analysis, and experimentation of chaos in a switched reluctance (SR) drive system using voltage pulsewidth modulation are presented. Based on the proposed nonlinear flux linkage model of the SR drive system, the computation time to evaluate the Poincaré map and its Jacobian matrix can be significantly shortened. Moreover, the stability analysis of the fundamental operation is conducted, leading to determine the stable parameter ranges and hence to avoid the occurrence of chaos. Both computer simulation and experimental measurement are given to verify the theoretical modeling and analysis.published_or_final_versio

Frequency-domain stability conditions for split-path nonlinear systems

This paper considers the class of control systems containing so-called split-path nonlinear (SPAN) filters, which are designed to overcome some of the well-known fundamental limitations in linear time-invariant (LTI) control. In this work, we are interested in developing tools for the stability analysis of such systems using frequency-domain techniques. Hereto, we explicitly show the equivalence between a set of linear matrix inequalities (LMIs) with S-procedure terms, guaranteeing stability of the closed-loop (SPAN) system, and a frequency-domain condition. We also provide a systematic procedure for verifying the frequency-domain condition in a graphical manner. The results are illustrated through a nummerical case study.</p