Related papers: On Dynamical Systems in the \'Etale Topology
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm…
The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this…
We introduce perfect resolving algebras and study their fundamental properties. These algebras are basic for our theory of differential graded schemes, as they give rise to affine differential graded schemes. We also introduce etale…
Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This…
We express some basic properties of Deninger's conjectural dynamical system in terms of morphisms of topoi. Then we show that the current definition of the Weil-\'etale topos satisfies these properties. In particular, the flow, the closed…
We present several topics involving the computation of dynamical systems. The emphasis is on work in progress and the presentation is informal -- there are many technical details which are not fully discussed. The topics are chosen to…
We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form,…
We give a description of the link between topological dynamical systems and their dimension groups. The focus is on minimal systems and, in particular, on substitution shifts. We describe in detail the various classes of systems including…
Electric circuits are usually described by charge- and flux-oriented modified nodal analysis. In this paper, we derive models as port-Hamiltonian systems on several levels: overall systems, multiply coupled systems and systems within…
A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as `machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special…
We investigate various topological spaces and varieties which can be associated to a block of a finite group scheme G. These spaces come from the theory of cohomological support varieties for modules, as well as from the…
This article explores some geometric and algebraic properties of the dynamical system which is represented by matrix differential equations arising from inertial navigation problems, such as the symplecticity and the orthogonality.…
We develop categorical foundations of discrete dynamical systems, aimed at understanding how the structure of the system affects its dynamics. The key technical innovation is the notion of a cycle set, which provides a formal language in…
The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous…
Synthetic algebraic geometry uses homotopy type theory extended with three axioms to develop algebraic geometry internal to a higher version of the Zariski topos. In this article we make no essential use of the higher structure and use…
In this article, we study the behaviour of discrete one-dimensional dynamical systems associated to functions on finite sets. We formalise the global orbit pattern formed by all the periodic orbits (gop) as the ordered set of periods when…
A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a…
We discuss some of the issues that arise in attempts to classify automorphisms of compact abelian groups from a dynamical point of view. In the particular case of automorphisms of one-dimensional solenoids, a complete description is given…
The questions of global topological, smooth and holomorphic classifications of the differential systems, defined by covering foliations, are considered. The received results are applied to nonautonomous linear differential systems and…
For any dynamical system $T:X\rightarrow X$ of a compact metric space $X$ with $g-$almost product property and uniform separation property, under the assumptions that the periodic points are dense in $X$ and the periodic measures are dense…