scholarly journals On the Rank and Periodic Rank of Finite Dynamical Systems

10.37236/7017 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Maximilien Gadouleau
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Points ◽  
Boolean Networks ◽  
Finite Alphabet ◽  
Interaction Graph ◽  
Maximum Rank ◽  
Local Functions ◽  
Finite Dynamical Systems ◽  
Finite Dynamical System

A finite dynamical system is a function $f : A^n \to A^n$ where $A$ is a finite alphabet, used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynamical system with a given interaction graph over any non-Boolean alphabet. The rank and the maximum rank are both computable in polynomial time. We also obtain a similar result for Boolean finite dynamical systems (also known as Boolean networks) whose interaction graphs are contained in a given digraph. We then prove that the average rank is relatively close (as the size of the alphabet is large) to the maximum. The results mentioned above only deal with the parallel update schedule. We finally determine the maximum rank over all block-sequential update schedules and the supremum periodic rank over all complete update schedules.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

On Devaney chaotic generalized shift dynamical systems

2013 ◽  
Vol 50 (4) ◽  
pp. 509-522 ◽  
Author(s):  
Fatemah Shirazi ◽  
Javad Sarkooh ◽  
Bahman Taherkhani
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Point ◽  
Periodic Points ◽  
List Type ◽  
Topologically Transitive ◽  
Generalized Shift ◽  
One To One ◽  
Shift Dynamical System ◽  
Infinite Sequences

In the following text we prove that in a generalized shift dynamical system (XГ, σφ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent: the dynamical system (XГ, σφ) is chaotic in the sense of Devaneythe dynamical system (XГ, σφ) is topologically transitivethe map φ: Г → Г is one to one without any periodic point.Also for infinite countable Г and finite discrete X with at least two elements (XГ, σφ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

Zero-dimensional covers of finite dimensional dynamical systems

1995 ◽  
Vol 15 (5) ◽  
pp. 939-950 ◽  
Author(s):  
John Kulesza
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Points ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

AbstractIf (X, f) is a compact metric, finite-dimensional dynamical system with a zero-dimensional set of periodic points, then there is a zero-dimensional compact metric dynamical system (C, g) and a finite-to-one (in fact, at most (n + l)n-to-one) surjection h: C → X such that h o g = f o h. An example shows that the requirement on the set of periodic points is necessary.

scholarly journals Shift-Induced Dynamical Systems on Partitions and Compositions

10.37236/1106 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brian Hopkins ◽  
Michael A. Jones
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Primary Result ◽  
Garden Of Eden ◽  
Finite Dynamical System

The rules of "Bulgarian solitaire" are considered as an operation on the set of partitions to induce a finite dynamical system. We focus on partitions with no preimage under this operation, known as Garden of Eden points, and their relation to the partitions that are in cycles. These are the partitions of interest, as we show that starting from the Garden of Eden points leads through the entire dynamical system to all cycle partitions. A primary result concerns the number of Garden of Eden partitions (the number of cycle partitions is known from Brandt). The same operation and questions can be put in the context of compositions (ordered partitions), where we give stronger results.

ON ASYNCHRONOUS CELLULAR AUTOMATA

2005 ◽  
Vol 08 (04) ◽  
pp. 521-538 ◽  
Author(s):  
A. Å. HANSSON ◽  
H. S. MORTVEIT ◽  
C. M. REIDYS
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Three Dimensional ◽  
Periodic Points ◽  
The State ◽  
Asynchronous Cellular Automata ◽  
Local Functions ◽  
Vertex Set ◽  
Dimensional Cube ◽  
A Finite Group

We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v1,…,vn} and where each vertex vi has a state xvi in a finite field K, (b) a sequence of functions (Fvi,Y)i, and (c) a word w = (w1,…,wk), where each wi is a vertex in Y. The function Fvi,Y updates the state of vertex vi as a function of the state of vi and its Y-neighbors and maps all other vertex states identically. The SDS is the composed map [Formula: see text]. In the particular case of ACA, the graph is the circle graph on n vertices (Y = Circ n), and all the maps Fvi are induced by a common Boolean function. Our main result is the identification of all w-independent ACA, that is, all ACA with periodic points that are independent of the word (update schedule) w. In general, for each w-independent SDS, there is a finite group whose structure contains information about for example SDS with specific phase space properties. We classify and enumerate the set of periodic points for all w-independent ACA, and we also compute their associated groups in the case of Y = Circ 4. Finally, we analyze invertible ACA and offer an interpretation of S35 as the group of an SDS over the three-dimensional cube with local functions induced by nor3 + nand3.

scholarly journals Dynamical properties of spatial discretizations of a generic homeomorphism

2014 ◽  
Vol 35 (5) ◽  
pp. 1474-1523 ◽  
Author(s):  
PIERRE-ANTOINE GUIHÉNEUF
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Numerical Study ◽  
Periodic Points ◽  
Dynamical Behaviour ◽  
Compact Manifolds ◽  
Spatial Discretization ◽  
Dynamical Features ◽  
Dynamical Properties

This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works on well-chosen examples (such as Gambaudo and Tresser [Some difficulties generated by small sinks in the numerical study of dynamical systems: two examples. Phys. Lett. A 94(9) (1983), 412–414]) show that the dynamical behaviours of dynamical systems and of their discretizations can be quite different. We are interested in generic homeomorphisms of compact manifolds. So our aim is to tackle the following question: can the dynamical properties of a generic homeomorphism be detected on the spatial discretizations of this homeomorphism? We will prove that the dynamics of a single discretization of a generic conservative homeomorphism does not depend on the homeomorphism itself, but rather on the grid used for the discretization. Therefore, dynamical properties of a given generic conservative homeomorphism cannot be detected using a single discretization. Nevertheless, we will also prove that some dynamical features of a generic conservative homeomorphism (such as the set of the periods of all periodic points) can be read on a sequence of finer and finer discretizations.

scholarly journals Parallel Dynamical Systems over Graphs and Related Topics: A Survey

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Juan A. Aledo ◽  
Silvia Martinez ◽  
Jose C. Valverde
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Evolution Operator ◽  
Discrete Dynamical System ◽  
Dynamical Behavior ◽  
Global Evolution ◽  
Local Functions

In discrete processes, as computational or genetic ones, there are many entities and each entity has a state at a given time. The update of states of the entities constitutes an evolution in time of the system, that is, a discrete dynamical system. The relations among entities are usually represented by a graph. The update of the states is determined by the relations of the entities and some local functions which together constitute (global) evolution operator of the dynamical system. If the states of the entities are updated in a synchronous manner, the system is called aparallel dynamical system. This paper is devoted to review the main results on the dynamical behavior of parallel dynamical systems over graphs which constitute a generic tool for modeling discrete processes.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Sign in / Sign up

Export Citation Format

Share Document