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 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.

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 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 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

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.

Various Shadowing in Linear Dynamical Systems

2019 ◽  
Vol 29 (03) ◽  
pp. 1950042 ◽  
Author(s):  
Xinxing Wu ◽  
Xu Zhang ◽  
Xin Ma
Keyword(s):  
Dynamical Systems ◽  
Linear Transformation ◽  
Positive Answer ◽  
Transformation Formula ◽  
Nonsingular Matrix ◽  
Linear Dynamical Systems ◽  
Shadowing Property ◽  
Matrix Formula ◽  
Asymptotic Average Shadowing

This paper proves that the linear transformation [Formula: see text] on [Formula: see text] has the (asymptotic) average shadowing property if and only if [Formula: see text] is hyperbolic, where [Formula: see text] is a nonsingular matrix, giving a positive answer to a question in [Lee, 2012a]. Besides, it is proved that [Formula: see text] does not have the [Formula: see text]-shadowing property, thus does not have the ergodic shadowing property for every nonsingular matrix [Formula: see text].

On the Dynamical Systems Stability Control and Applications

2014 ◽  
Vol 555 ◽  
pp. 361-368
Author(s):  
Marcel Migdalovici ◽  
Daniela Baran ◽  
Gabriela Vlădeanu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear System ◽  
Stability Control ◽  
Necessary Condition ◽  
Linear Dynamical Systems ◽  
System A ◽  
First Approximation ◽  
Free Parameters ◽  
The Stability

The stability control analyzed by us, in this show, is based on our results in the domain of dynamical systems that depend of parameters. Any dynamical system can be considered as dynamical system that depends of parameters, without numerical particularization of them. All concrete dynamical systems, meted in the specialized literature, underline the property of separation between the stable and unstable zones, in sense of Liapunov, for two free parameters. This property can be also seen for one or more free parameters. Some mathematical conditions of separation between stable and unstable zones for linear dynamical systems are identified by us. For nonlinear systems, the conditions of separation may be identified using the linear system of first approximation attached to nonlinear system. A necessary condition of separation between stable and unstable zones, identified by us, is the sufficient order of differentiability or conditions of continuity for the functions that define the dynamical system. The property of stability zones separation can be used in defining the strategy of stability assurance and optimizing of the parameters, in the manner developed in the paper. The cases of dynamical systems that assure the separations of the stable and unstable zones, in your evolution, and permit the stability control, are analyzed in the paper.

Optimization of a Certain Quality of Complete Controllability and Observability for Linear Dynamical Systems

1969 ◽  
Vol 91 (2) ◽  
pp. 228-237 ◽  
Author(s):  
C. D. Johnson
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Structural Parameters ◽  
Computational Procedure ◽  
Complete Controllability ◽  
Linear Dynamical Systems ◽  
Controllability And Observability ◽  
Meaningful Measure

The problem of assigning a physically meaningful measure of the “quality” of controllability and observability to dynamical systems which are completely controllable and/or completely observable is considered. One particular analytical measure of quality, for a class of linear dynamical systems, is proposed and an effective computational procedure for maximizing the proposed quality, with respect to certain adjustable structural parameters of the dynamical system, is described. Three illustrative examples are worked in detail.

scholarly journals VIRUS AND WARNING SPREAD IN DYNAMICAL NETWORKS

2011 ◽  
Vol 14 (03) ◽  
pp. 341-358 ◽  
Author(s):  
CARLOS RODRIGUEZ LUCATERO ◽  
ROBERTO BERNAL JAQUEZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
System Approach ◽  
P2p Networks ◽  
System Matrix ◽  
Linear Dynamical Systems ◽  
Warning Messages ◽  
Stability Theorems ◽  
Virus Spreading ◽  
The Web

Recent work on information survival in sensor and human P2P networks try to study the datum preservation or the virus spreading in a network under the dynamical system approach. Some interesting solutions propose to use non-linear dynamical systems and fixed point stability theorems, providing closed form formulas that depend on the largest eigenvalue of the dynamic system matrix. Given that in the Web there can be messages from one place to another, and that these messages can be, with some probability, new unclassified virus warning messages as well as worms or other kind of viruses, the sites can be infected very fast. The question to answer is how and when a network infection can become global and how it can be controlled or at least how to stabilize the spreading in such a way that it becomes confined below a fixed portion of the network. In this paper, we try to be a step ahead in this direction and apply classic results of the dynamical systems theory to model the behavior of a network where warning messages and viruses spread.

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