scholarly journals Lifespan in a Primitive Boolean Linear Dynamical System

10.37236/5193 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Yaokun Wu ◽  
Yinfeng Zhu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Dynamical Systems ◽  
Minimum Size ◽  
Nonnegative Matrices ◽  
Linear Dynamical System ◽  
Linear Dynamical Systems

Let $\mathcal F$ be  a set of $k$ by $k$ nonnegative matrices such that every "long" product of elements of $\mathcal F$ is positive.   Cohen and Sellers (1982) proved that, then,  every such product of length $2^k-2$ over $\mathcal F$ must be positive. They suggested to investigate the minimum size of such $\mathcal F$ for which there exists a non-positive  product of length $2^k-3$ over $\mathcal F$ and they constructed one example of size $2^k-2$.  We construct one of size $k$ and further discuss relevant basic problems in the framework of Boolean linear dynamical systems. We also formulate several primitivity properties for general discrete dynamical systems.

scholarly journals STABILITY OF NON-LINEAR DYNAMICAL SYSTEM

2021 ◽  
Vol 9 (07) ◽  
pp. 275-283
Author(s):  
Anas Salim Youns ◽  
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Linear System ◽  
Three Dimension ◽  
Linear Dynamical System ◽  
Linear Dynamical Systems ◽  
Linearization Technique ◽  
The Public ◽  
Non Linear ◽  
The Stability

The mainobjective of this research is to study the stability of thenon-lineardynamical system by using the linearization technique of three dimension systems toobtain an approximate linear system and find its stability. We apply this technique to reaches to the stability of the public non linear dynamical systems of dimension. Finally, some proposed examples (example (1) and example (2)) are given to explain this technique and used the corollary.

Linear dynamical systems of dimension two over the ring of integers modulo pt

2020 ◽  
Vol 12 (06) ◽  
pp. 2050074
Author(s):  
Yangjiang Wei ◽  
Heyan Xu ◽  
Linhua Liang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Diagonal Matrix ◽  
Linear Dynamical System ◽  
Linear Dynamical Systems ◽  
Ring Of Integers ◽  
Idempotent Matrix ◽  
Nilpotent Matrix ◽  
Matrix Formula

In this paper, we investigate the linear dynamical system [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text] ([Formula: see text] is a prime). In order to facilitate the visualization of this system, we associate a graph [Formula: see text] on it, whose nodes are the points of [Formula: see text], and for which there is an arrow from [Formula: see text] to [Formula: see text], when [Formula: see text] for a fixed [Formula: see text] matrix [Formula: see text]. In this paper, the in-degree of each node in [Formula: see text] is obtained, and a complete description of [Formula: see text] is given, when [Formula: see text] is an idempotent matrix, or a nilpotent matrix, or a diagonal matrix. The results in this paper generalize Elspas’ [1959] and Toledo’s [2005].

scholarly journals Reflections on Termination of Linear Loops

2021 ◽  
pp. 51-74
Author(s):  
Shaowei Zhu ◽  
Zachary Kincaid
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Transition System ◽  
Linear Dynamical System ◽  
Linear Dynamical Systems ◽  
Termination Analysis ◽  
Integer Arithmetic ◽  
Transition Formula

AbstractThis paper shows how techniques for linear dynamical systems can be used to reason about the behavior of general loops. We present two main results. First, we show that every loop that can be expressed as a transition formula in linear integer arithmetic has a best model as a deterministic affine transition system. Second, we show that for any linear dynamical system f with integer eigenvalues and any integer arithmetic formula G, there is a linear integer arithmetic formula that holds exactly for the states of f for which G is eventually invariant. Combining the two, we develop a monotone conditional termination analysis for general loops.

scholarly journals Designing the mesoscopic approach of an autonomous linear dynamical system by a quantum formulation

2016 ◽  
Author(s):  
Juan Carlos Micó Ruiz
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Linear Systems ◽  
Second Order ◽  
Linear Dynamical System ◽  
Linear Dynamical Systems ◽  
General Systems ◽  
Non Linear ◽  
Quantum Formulation

The work presents a mesoscopic approach to general systems modelled by dynamical systems. The quantum formulation is possible to be obtained by their quantum formulation from a second order Hamiltonian. However, only autonomous linear systems are proved to obtain a Hamiltonian like this. Some application cases are presented, and a discussion about how to generalize the formalism to non-linear dynamical systems is sketched.DOI: http://dx.doi.org/10.4995/IFDP.2016.2795

scholarly journals Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems

2019 ◽  
Vol 24 (1) ◽  
pp. 13 ◽  
Author(s):  
Francisco Solis
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Dynamical Systems ◽  
Dynamical Behavior ◽  
Discrete Dynamical Systems ◽  
Exponential Polynomial ◽  
Linear Dynamical System ◽  
The Stability ◽  
Polynomial Family ◽  
Complex Dynamical Behavior

In this paper, we introduce and analyze a family of exponential polynomial discrete dynamical systems that can be considered as functional perturbations of a linear dynamical system. The stability analysis of equilibria of this family is performed by considering three different parametric scenarios, from which we show the intricate and complex dynamical behavior of their orbits.

scholarly journals Chaotic Dynamical Systems Associated with Tilings of RN

2011 ◽  
pp. 19-41
Author(s):  
Lionel Rosier
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Homogeneous Space ◽  
Chaos Synchronization ◽  
Discrete Dynamical Systems ◽  
Dimensional Torus ◽  
Synchronization Mechanism ◽  
Chaotic Dynamical Systems ◽  
Regular Tiling

In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of RN, whose most familiar example is provided by the N-dimensional torus TN. It is proved that any dynamical system in this class is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent. Next, a chaos-synchronization mechanism is introduced and used for masking information in a communication setup.

MODELING LINEAR DYNAMICAL SYSTEMS BY CONTINUOUS-VALUED CELLULAR AUTOMATA

2007 ◽  
Vol 18 (05) ◽  
pp. 833-848 ◽  
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
MANUEL GONZALEZ HERNANDEZ ◽  
NORBERTO HERNANDEZ ROMERO ◽  
AARON RODRIGUEZ TREJO ◽  
SERGIO V. CHAPA VERGARA
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Linear Systems ◽  
Configuration Space ◽  
Integration Method ◽  
Numerical Calculations ◽  
Linear Dynamical Systems ◽  
Original Part

This paper exposes a procedure for modeling and solving linear systems using continuous-valued cellular automata. The original part of this work consists on showing how the cells in the automaton may contain both real values and operators for carrying out numerical calculations and solve a desired problem. In this sense the automaton acts as a program, where data and operators are mixed in the evolution space for obtaining the correct calculations. As an example, Euler's integration method is implemented in the configuration space in order to achieve an approximated solution for a dynamical system. Three examples showing linear behaviors are presented.

scholarly journals A unified approach for sparse dynamical system inference from temporal measurements

2018 ◽  
Vol 35 (18) ◽  
pp. 3387-3396 ◽  
Author(s):  
Yannis Pantazis ◽  
Ioannis Tsamardinos
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Single Cell ◽  
Time Course ◽  
Cell Mass ◽  
Supplementary Information ◽  
Unified Approach ◽  
Unknown Parameters ◽  
Inference Problem ◽  
Linear Dynamical Systems

Abstract Motivation Temporal variations in biological systems and more generally in natural sciences are typically modeled as a set of ordinary, partial or stochastic differential or difference equations. Algorithms for learning the structure and the parameters of a dynamical system are distinguished based on whether time is discrete or continuous, observations are time-series or time-course and whether the system is deterministic or stochastic, however, there is no approach able to handle the various types of dynamical systems simultaneously. Results In this paper, we present a unified approach to infer both the structure and the parameters of non-linear dynamical systems of any type under the restriction of being linear with respect to the unknown parameters. Our approach, which is named Unified Sparse Dynamics Learning (USDL), constitutes of two steps. First, an atemporal system of equations is derived through the application of the weak formulation. Then, assuming a sparse representation for the dynamical system, we show that the inference problem can be expressed as a sparse signal recovery problem, allowing the application of an extensive body of algorithms and theoretical results. Results on simulated data demonstrate the efficacy and superiority of the USDL algorithm under multiple interventions and/or stochasticity. Additionally, USDL’s accuracy significantly correlates with theoretical metrics such as the exact recovery coefficient. On real single-cell data, the proposed approach is able to induce high-confidence subgraphs of the signaling pathway. Availability and implementation Source code is available at Bioinformatics online. USDL algorithm has been also integrated in SCENERY (http://scenery.csd.uoc.gr/); an online tool for single-cell mass cytometry analytics. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals Discrete Dynamical Systems: A Brief Survey

2018 ◽  
Vol 14 (1) ◽  
pp. 35-51
Author(s):  
Sara Fernandes ◽  
Carlos Ramos ◽  
Gyan Bahadur Thapa ◽  
Luís Lopes ◽  
Clara Grácio
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Complex Numbers ◽  
Survey Paper ◽  
Number Systems ◽  
Mathematical Formalization ◽  
Research Students ◽  
Work Done

 Dynamical system is a mathematical formalization for any fixed rule that is described in time dependent fashion. The time can be measured by either of the number systems - integers, real numbers, complex numbers. A discrete dynamical system is a dynamical system whose state evolves over a state space in discrete time steps according to a fixed rule. This brief survey paper is concerned with the part of the work done by José Sousa Ramos [2] and some of his research students. We present the general theory of discrete dynamical systems and present results from applications to geometry, graph theory and synchronization. Journal of the Institute of Engineering, 2018, 14(1): 35-51

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