Dynamical Systems in Categories

Behrisch, Mike; Kerkhoff, Sebastian; Pöschel, Reinhard; Schneider, Friedrich Martin; Siegmund, Stefan

doi:10.1007/s10485-015-9409-8

Cited by 7 publications

(8 citation statements)

References 36 publications

(36 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For more background on monoidal categories and functors, refer [3] and [4]. Our approach is different from the one in [5]. e dynamical systems presented here are defined as a generalisation of automata whose input and output spaces are predetermined.…”

Section: Boxes and Wiring Diagramsmentioning

confidence: 99%

Memoryless Systems Generate the Class of all Discrete Systems

Beurier

Pastor

Spivak

2019

International Journal of Mathematics and Mathematical Sciences

View full text Add to dashboard Cite

Automata are machines, which receive inputs, accordingly update their internal state, and produce output, and are a common abstraction for the basic building blocks used in engineering and science to describe and design complex systems. These arbitrarily simple machines can be wired together—so that the output of one is passed to another as its input—to form more complex machines. Indeed, both modern computers and biological systems can be described in this way, as assemblies of transistors or assemblies of simple cells. The complexity is in the network, i.e., the connection patterns between simple machines. The main result of this paper is to show that the range of simplicity for parts as compared to the complexity for wholes is in some sense complete: the most complex automaton can be obtained by wiring together direct-output memoryless components. The model we use—discrete-time automata sending each other messages from a fixed set of possibilities—is certainly more appropriate for computer systems than for biological systems. However, the result leads one to wonder what might be the simplest sort of machines, broadly construed, that can be assembled to produce the behaviour found in biological systems, including the brain.

show abstract

Section: Boxes and Wiring Diagramsmentioning

confidence: 99%

Memoryless Systems Generate the Class of all Discrete Systems

Beurier

Pastor

Spivak

2019

International Journal of Mathematics and Mathematical Sciences

View full text Add to dashboard Cite

show abstract

“…In this paper, we aim at revisiting and further developing the category theory-based modelling methodology introduced in [13]. The motivation for using category theory for abstract description of mathematical models is based on several aspects: (i) The abstract nature of category theory allows description of very different objects and structures on common basis; (ii) a practical interpretation of abstract constructions provided by category theory-based modelling methodology is straightforward, and thus the methodology can really be used in engineering practice; (iii) category theory naturally provides scaling possibilities implying that description of more sophisticated objects and structures can be done by using the same principles as descriptions of their individual parts; (iv) finally, various applications of category theory scattering from modelling of dynamical systems [14] to ontological representation of knowledge [15] presented in recent years indicate that advantages of category theory are seen and accepted now not only by mathematicians, but also by people interested in applications.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Modelling by Help of Category Theory: Models and Relations between Them

Legatiuk

2021

Mathematics

View full text Add to dashboard Cite

The growing complexity of modern practical problems puts high demand on mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice is becoming particularly important. Methods for model comparison and model choice typically used in practical applications nowadays are computation-based, and thus time consuming and computationally costly. Therefore, it is necessary to develop other approaches to working abstractly, i.e., without computations, with mathematical models. An abstract description of mathematical models can be achieved by the help of abstract mathematics, implying formalisation of models and relations between them. In this paper, a category theory-based approach to mathematical modelling is proposed. In this way, mathematical models are formalised in the language of categories, relations between the models are formally defined and several practically relevant properties are introduced on the level of categories. Finally, an illustrative example is presented, underlying how the category-theory based approach can be used in practice. Further, all constructions presented in this paper are also discussed from a modelling point of view by making explicit the link to concrete modelling scenarios.

show abstract

“…Thus, it will be possible to determine several quantities, such as the Lyapunov exponents, that will give us rich in-formation about the systems dynamical behavior [14]. It is worth noticing that the abstract theory of dynamical systems in the context of categories has been already considered by diverse authors but, within a pure theoretic spirit, as a tool to relate different mathematical fields and concepts [15,16].…”

Section: Introductionmentioning

confidence: 99%

Hierarchical evolutive systems, fuzzy categories and the living single cell

Mesón¹,

Carlevaro²,

Vericat³

2018

Preprint

View full text Add to dashboard Cite

In this article, the theory of hierarchical evolutive systems of Ehresmann and Vandremeersch [Bull. Math. Bio. 49, 13-50 (1987)] is improved by considering the categories of the theory as fuzzy sets whose elements are the composite objects formed by the arrows and corresponding vertices of their embedded graphs. This way each category can be represented as a point in the states space [0, 1]N ⊂ R N . The introduction of a diffeomorphism, that acts in this context as a functor between categories, allows to define a measure preserving dynamical system. In particular, we apply this formalism to describe a living single cell. We propose for its state at a given time a hirerchical category with three levels (molecular, coarse-grained and cellular levels) related by adequate colimits. Each level involves the main functional and structural modules in which the cell can be partitioned. The time evolution of the cell is drived by a transformation which is a N -dimensional generalization of the Ricker map whose parameters we propose to be determined by requiring that, as hallmark of its behavior, the living cell to evolve at the edge of chaos. From the dynamical point of view this property manifests in the fact that the largest Lyapunov

show abstract

Dynamical Systems in Categories

Cited by 7 publications

References 36 publications

Memoryless Systems Generate the Class of all Discrete Systems

Memoryless Systems Generate the Class of all Discrete Systems

Mathematical Modelling by Help of Category Theory: Models and Relations between Them

Hierarchical evolutive systems, fuzzy categories and the living single cell

Contact Info

Product

Resources

About