Dynamical Systems and Sheaves

Abstract

A. 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 cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally, morphisms between the different types of systems allow their translation in a common framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly of different types, as well as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.

Operadic Modeling of Dynamical Systems: Mathematics and Computation

“…Two new algebras for composing dynamical systems. The first algebra represents a directed composition of continuous and discrete systems that extends the syntax of algebras previously studied in [18,15,21] to include merging and creating wires. The second algebra represents the undirected composition of discrete dynamical systems.…”

“…In [21,15,7] the directed syntax for composing dynamical systems is defined by the operad underlying Lens FinSet op , +, 0 0 , often referred to as the operad of wiring diagrams. Following the Catlab implementation, we instead focus on operad underlying Lens Cospan(FinSet op ) , +, 0 0 .…”

A Compositional Sheaf-Theoretic Framework for Event-Based Systems

“…CPSs are complex systems including physical components, such as actuators, sensors, and computer units, and software components, such as perception, planning, and control modules. Starting from the first mention of CPSs, their formal description has been object of studies in several disciplines, such as control theory [2,20,5,7], computer science [8,16,15,10,11], and applied category theory [1,17,18].…”

Amplified steady state bifurcations in feedforward networks

“…More recently, so-called open systems (see e.g. [15,27]) and asynchronous networks ( [3,4,5]) have been introduced to model more complex applications. Network dynamical systems exhibit interesting phenomena that are highly anomalous in general dynamical systems.…”