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Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of…
Using tools from computable analysis we develop a notion of effectiveness for general dynamical systems as those group actions on arbitrary spaces that contain a computable representative in their topological conjugacy class. Most natural…
Discontinuous dynamical systems with grazing solutions are discussed. The group property, continuation of solutions, continuity and smoothness of motions are thoroughly analyzed. A variational system around a grazing solution which depends…
This paper proposes a systems approach to social sciences based on mathematical framework derived from a generalization of the mathematical kinetic theory and on theoretical tools of game theory. Social systems are modeled as a living…
The dissipation function for a system is defined as the natural logarithm of the ratio between probabilities of a trajectory and its time-reversed trajectory, and its probability distribution follows a well-known relation called the…
We define the infinite dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^infinity, and prove that this space has the 'fixed point property': any continuous function from the space into itself has a…
The present communication addresses a set of observations, obeying both deterministic as well as statistical formal requirements, and serving to operate within the framework of the dynamical systems theory, with a certain emphasis placed on…
The theory of complex networks and of disordered systems is used to study the stability and dynamical properties of a simple model of material flow networks defined on random graphs. In particular we address instabilities that are…
This paper draws distinctions among various concepts related to tipping points, robustness, path dependence, and other properties of system dynamics. For each concept a formal definition is provided that utilizes Markov model…
Understanding why models behave the way they do is critical to learning from them, and to conveying the insights they offer to a broad audience. The Loops that Matter methodology automatically shows which loops are dominating behavior at…
The beauty of physics is that there is usually a conserved quantity in an always-changing system, known as the constant of motion. Finding the constant of motion is important in understanding the dynamics of the system, but typically…
We consider a generic nonlinear extension of May's 1972 model by including all higher-order terms in the expansion around the chosen fixed point (placed at the origin) with random Gaussian coefficients. The ensuing analysis reveals that as…
It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. The existing definitions of chaos are formulated in sets of motions. This is…
We study the topology of circularly ordered sets. While the algebraic notion is classical, the general topological theory has received comparatively little attention. In this work we provide a self-contained topological exposition and…
Model sets are projections of certain lattice subsets. It was realised by Moody that dynamical properties of such sets are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the…
Hegselmann--Krause models are localized, distributed averaging dynamics on spatial data. A key aspect of these dynamics is that they lead to cluster formation, which has important applications in geographic information systems, dynamic…
The advent and proliferation of social media have led to the development of mathematical models describing the evolution of beliefs/opinions in an ecosystem composed of socially interacting users. The goal is to gain insights into…
Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for…
In recent years, several global events have severely disrupted economies and social structures, undermining confidence in the resilience of modern societies. While empirical evidence on the dynamics and drivers of past societal collapse is…
Robustness guarantees are important properties to be looked for during control design. They ensure stability of closed-loop systems in face of uncertainties, unmodeled effects and bounded disturbances. While the theory on robust stability…