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We introduce dynamical systems $(X,G)$ with closed relations $G$ on compact metric spaces $X$ and discuss different types of minimality of such dynamical systems, all of them generalizing minimal dynamical systems $(X,f)$ with continuous…
Many mechanical systems exhibit changes in their kinematic topology altering the mobility. Ideal contact is the best known cause, but also stiction and controlled locking of parts of a mechanism lead to topology changes. The latter is…
We develop a taxonomy of different behavioral specification theories and expose their algebraic properties. We start by clarifying what precisely constitutes a behavioral specification theory and then introduce logical and structural…
The Cahn-Hilliard equation and extensions, notably the Cahn-Hilliard-Darcy and Cahn-Hilliard-Navier-Stokes systems, provide widely used frameworks for coupling interfacial thermodynamics with flow. This review surveys the thermodynamic…
We consider characterisations of unitary dilations and approximations of irreversible classical dynamical systems on a Hilbert space. In the commutative case, building on the work in [9], one can express well known approximants (e.g. Hille-…
We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary…
We describe the approximation of a continuous dynamical system on a p. l. manifold or Cantor set by a tractable system. A system is tractable when it has a finite number of chain components and, with respect to a given full background…
Continuous-time projected dynamical systems are an elementary class of discontinuous dynamical systems with trajectories that remain in a feasible domain by means of projecting outward-pointing vector fields. They are essential when…
In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For…
Modeling a sequence of design steps, or a sequence of parameter settings, yields a sequence of dynamical systems. In many cases, such a sequence is intended to approximate a certain limit case. However, formally defining that limit turns…
We define the notion of localizable property for a dynamical system. Then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems. These notions are the fast growth of the…
In order to investigate the evolutionary process of many deterministic Dynamical systems with unfixed parameter, a set of dynamical models with parameter changing continuously and the accumulation of this change might be large is introduced…
There are three important types of structural properties that remain unchanged under the structural transformation of condensed matter physics and chemistry. They are the properties that remain unchanged under the structural periodic…
Reservoir computing systems are constructed using a driven dynamical system in which external inputs can alter the evolving states of a system. These paradigms are used in information processing, machine learning, and computation. A…
We consider Markov decision processes (MDPs) which are a standard model for probabilistic systems. We focus on qualitative properties for MDPs that can express that desired behaviors of the system arise almost-surely (with probability 1) or…
The literature on concurrency theory offers a wealth of examples of characteristic-formula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed…
This paper is motivated by the theory of sequential dynamical systems, developed as a basis for a mathematical theory of computer simulation. It contains a classification of finite dynamical systems on binary strings, which are obtained by…
This paper introduces improved numerical techniques for addressing numerical boundary and interface coupling conditions in the context of diffusion equations in cellular biophysics or heat conduction problems in fluid-structure…
In this work we construct the $\Co^{\r}$-completion and $\Co^{\l}$-completion of a dynamical system. If $X$ is a flow, we construct canonical maps $X\to \Co^{\r}(X)$ and $X\to \Co^{\l}(X)$ and when these maps are homeomorphism we have the…
We show that attractors are semicontinuous for closed relations on compact Hausdorff spaces. Semicontinuity is what guarantees that small changes to a system do not result in massive growth of certain features, notably attractors. That is,…