Reducing nonlinear dynamical systems to canonical forms

A Direct Entropic Approach to the Thermal Balance of Spontaneous Chemical Reactions

When working with, and learning about, the thermal balance of a chemical reaction, we need to consider two overlapping but conceptually distinct aspects: one relates to the process of reallocating entropy between reactants and products (because of different specific entropies of the new substances compared to those of the old), and the other to dissipative processes. Together, they determine how much entropy is exchanged between the chemicals and their environment (i.e., in heating and cooling). By making explicit use of (a) the two conjugate pairs chemical amount (i.e., amount of substance) and chemical potential, and entropy and temperature, respectively, (b) the laws of balance of amount of substance on the one hand and entropy on the other, and (c) a generalized approach to the energy principle, it is possible to create both imaginative and formal conceptual tools for modeling thermal balances associated with chemical transformations in general and exothermic and endothermic reactions in particular. In this paper, we outline the concepts and relations needed for a direct approach to chemical and thermal dynamics, create a model of exothermic and endothermic reactions, including numerical examples, and discuss how to relate the direct entropic approach to traditional models of these phenomena.

Endemic oscillations for SARS-CoV-2 Omicron-A SIRS model analysis

The SIRS model with constant vaccination and immunity waning rates is well known to show a transition from a disease-free to an endemic equilibrium as the basic reproduction number r0 is raised above threshold. It is shown that this model maps to Hethcote's classic endemic model originally published in 1973. In this way one obtains unifying formulas for a whole class of models showing endemic bifurcation. In particular, if the vaccination rate is smaller than the recovery rate and r-<r0<r+ for certain upper and lower bounds r±, then trajectories spiral into the endemic equilibrium via damped infection waves. Latest data of the SARS-CoV-2 Omicron variant suggest that according to this simplified model continuous vaccination programs will not be capable to escape the oscillating endemic phase. However, in view of the strong damping factors predicted by the model, in reality these oscillations will certainly be overruled by time-dependent contact behaviors.

Data-Driven Corrections of Partial Lotka-Volterra Models

In many applications of interacting systems, we are only interested in the dynamic behavior of a subset of all possible active species. For example, this is true in combustion models (many transient chemical species are not of interest in a given reaction) and in epidemiological models (only certain subpopulations are consequential). Thus, it is common to use greatly reduced or partial models in which only the interactions among the species of interest are known. In this work, we explore the use of an embedded, sparse, and data-driven discrepancy operator to augment these partial interaction models. Preliminary results show that the model error caused by severe reductions-e.g., elimination of hundreds of terms-can be captured with sparse operators, built with only a small fraction of that number. The operator is embedded within the differential equations of the model, which allows the action of the operator to be interpretable. Moreover, it is constrained by available physical information and calibrated over many scenarios. These qualities of the discrepancy model-interpretability, physical consistency, and robustness to different scenarios-are intended to support reliable predictions under extrapolative conditions.