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We propose a dynamical model for group formation and switching behavior in systems where each group competes for members through attraction functions that are inversely proportional to their current sizes. This attraction is modulated by…
A discrete-time version of the replicator equation for two-strategy games is studied. The stationary properties differ from that of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replace…
We consider both the dynamics within and towards the supercycle attractors along the period-doubling route to chaos to analyze the development of a statistical-mechanical structure. In this structure the partition function consists of the…
We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex…
During modeling of dynamical systems, often two or more model architectures are combined to obtain a more powerful or efficient model regarding a specific application area. This covers the combination of multiple machine learning…
Many engineered as well as naturally occurring dynamical systems do not have an accurate mathematical model to describe their dynamic behavior. However, in many applications, it is possible to probe the system with external inputs and…
Patterns embody repeating phenomena, and, as such, they are partly but not fully detachable from their context. 'Design patterns' and 'pattern languages' are established methods for working with patterns. They have been applied in…
The aim of this paper is to describe the structure of global attractors for non-autonomous difference systems of equations with recurrent (in particular, almost periodic) coefficients. We consider a special class of this type of systems…
This paper refined and introduced some notations (namely attractors, physical attractors, proper attractors, topologically exact and topologically mixing) within the context of relations. We establish necessary and sufficient conditions,…
This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik and is based on a number of mini-courses taught by the author at University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern…
In applied sciences, such as physics and biology, it is often required to model the evolution of populations via dynamical systems. In this paper, we focus on the problem of approximating the basins of attraction of such models in case of…
In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This problem is rather common especially in population dynamics models. Precisely, a particular solution of a…
A new method is proposed which allows a reconstruction of time series based on higher order multiscale statistics given by a hierarchical process. This method is able to model the time series not only on a specific scale but for a range of…
Proper states' representations are the key to the successful dynamics modeling of chaotic systems. Inspired by recent advances of deep representations in various areas such as natural language processing and computer vision, we propose the…
We review the statistical mechanics approach to the study of the emerging collective behavior of systems of heterogeneous interacting agents. The general framework is presented through examples is such contexts as ecosystem dynamics and…
Several real-world and abstract structures and systems are characterized by marked hierarchy to the point of being expressed as trees. Because the study of these entities often involves sampling (or discovering) the tree nodes in a specific…
Recent work in dynamical systems theory has shown that many properties that are associated with irreversible processes in fluids can be understood in terms of the dynamical properties of reversible, Hamiltonian systems. That is,…
When studying social, economic and biological systems, one has often access to only limited information about the structure of the underlying networks. An example of paramount importance is provided by financial systems: information on the…
In this paper, we provide an overview of the research conducted in the context of structural systems since the latest survey by Dion et al. in 2003. We systematically consider all the papers that cite this survey as well as the seminal work…
Traditional economic models typically treat private information, or signals, as generated from some underlying state. Recent work has explicated alternative models, where signals correspond to interpretations of available information. We…