scholarly journals On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1088 ◽  
Author(s):  
Juan A. Aledo ◽  
Ali Barzanouni ◽  
Ghazaleh Malekbala ◽  
Leila Sharifan ◽  
Jose C. Valverde
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Periodic Structure ◽  
Boolean Functions ◽  
Directed Graphs ◽  
General Pattern ◽  
General Situation ◽  
Point System ◽  
Local Functions

In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system.

scholarly journals Coexistence of Periods in Parallel and Sequential Boolean Graph Dynamical Systems over Directed Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1812
Author(s):  
Juan A. Aledo ◽  
Luis G. Diaz ◽  
Silvia Martinez ◽  
Jose C. Valverde
Keyword(s):  
Dynamical Systems ◽  
Boolean Function ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Periodic Structure ◽  
Directed Graphs ◽  
Dependency Graph ◽  
Evolution Operators ◽  
Undirected Graphs

In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show that periodic orbits of any period can coexist in both kinds of update schedules, parallel and sequential. This result contrasts with the properties of their counterparts over undirected graphs with the same evolution operators, where fixed points cannot coexist with periodic orbits of other different periods. These results complete the study of the periodic structure of homogeneous Boolean graph dynamical systems on maxterm and minterm functions.

scholarly journals Dynamics of Endomorphism of Finite Abelian Groups

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Zhao Jinxing ◽  
Nan Jizhu
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Dynamical Systems ◽  
Automorphism Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Point System ◽  
Finite Abelian Groups

We study the dynamics of endomorphisms on a finite abelian group. We obtain the automorphism group for these dynamical systems. We also give criteria and algorithms to determine whether it is a fixed point system.

scholarly journals On the Periods of Parallel Dynamical Systems

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Juan A. Aledo ◽  
Luis G. Diaz ◽  
Silvia Martinez ◽  
Jose C. Valverde
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Fixed Point Theorems ◽  
Evolution Operators ◽  
Global Evolution

In this work, we provide conditions to obtain fixed point theorems for parallel dynamical systems over graphs with (Boolean) maxterms and minterms as global evolution operators. In order to do that, we previously prove that periodic orbits of different periods cannot coexist, which implies that Sharkovsky’s order is not valid for this kind of dynamical systems.

PERIODICITY AND SHARKOVSKY'S THEOREM FOR RANDOM DYNAMICAL SYSTEMS

2001 ◽  
Vol 01 (03) ◽  
pp. 299-338 ◽  
Author(s):  
MARC KLÜNGER
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Fixed Point Theorem ◽  
Periodic Orbits ◽  
Point Theorem ◽  
Random Dynamical Systems ◽  
Random Fixed Point ◽  
One Dimensional ◽  
Sharkovsky's Theorem ◽  
Random Fixed Point Theorem

We generalize the deterministic notion of periodicity to random dynamical systems, which leads to three different objects, called random periodic orbits, point and cycles. We analyze the relation of these three notions and prove a "random fixed point theorem" for one-dimensional random dynamical systems. Finally we use these notions to prove partial generalizations of Sharkovsky's theorem to random dynamical systems.

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

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