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Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given…

Motivated by questions in mass-action kinetics, we introduce the notion of vertexical family of differential inclusions. Defined on open hypercubes, these families are characterized by particular good behavior under projection maps. The…

Previous theoretical, along with early simulation and experimental, studies have indicated that particles with a short-ranged attraction exhibit a range of new dynamical arrest phenomena. These include very pronounced reentrance in the…

Inferring network topology from dynamical observations is a fundamental problem pervading research on complex systems. Here, we present a simple, direct method to infer the structural connection topology of a network, given an observation…

In this paper, we introduce ergodic sets, subsets of nodes of the networks that are dynamically disjoint from the rest of the network (i.e. that can never be reached or left following to the network dynamics). We connect their definition to…

Data-driven approaches are increasingly popular for identifying dynamical systems due to improved accuracy and availability of sensor data. However, relying solely on data for identification does not guarantee that the identified systems…

Probabilistic circuits compute multilinear polynomials that represent multivariate probability distributions. They are tractable models that support efficient marginal inference. However, various polynomial semantics have been considered in…

This paper addresses the problem of model reduction for dynamical system models that describe biochemical reaction networks. Inherent in such models are properties such as stability, positivity and network structure. Ideally these…

We give a self-contained introduction to the theory of directed graphs, leading up to the relationship between the Perron-Frobenius eigenvectors of a graph and its autocatalytic sets. Then we discuss a particular dynamical system on a fixed…

This paper presents a framework for abstracting uncertain or non-polynomial components of dynamical systems using polynomial constraints. This enables the application of polynomial-based analysis tools, such as sum-of-squares programming,…

Thermodynamics-informed neural networks employ inductive biases for the enforcement of the first and second principles of thermodynamics. To construct these biases, a metriplectic evolution of the system is assumed. This provides excellent…

We consider toric dynamical systems, which are also called complex-balanced mass-action systems. These are remarkably stable polynomial dynamical systems that arise from the analysis of mathematical models of reaction networks when, under…

An efficient numerical framework is presented for modeling viscoelasticity and permanent set of polymers. It is based on the hereditary integral form of transient network theory, in which polymer chains belong to distinct networks each with…

We consider symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then…

In this paper we study a class of dynamical systems generated by iterations of multivariate permutation polynomial systems which lead to polynomial growth of the degrees of these iterations. Using these estimates and the same techniques…

Persistence and permanence are properties of dynamical systems that describe the long-term behavior of the solutions, and in particular specify whether positive solutions approach the boundary of the positive orthant. Mass-action systems…

Models of complex networks are generally defined as graph stochastic processes in which edges and vertices are added or deleted over time to simulate the evolution of networks. Here, we define a unifying framework - probabilistic inductive…

Certain classes of problems, including perceptual data understanding, robotics, discovery, and learning, can be represented as incremental, dynamically constructed belief networks. These automatically constructed networks can be dynamically…

We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{|E(G)|},…

The practical description of disordered chemical reactions, where reactions involve multiple species at multiple sites, is presently challenge using correlated electronic structure methods. Here we describe the gradient theory of…