scholarly journals Networks and dynamical systems

1993 ◽  
Vol 25 (1) ◽  
pp. 140-175 ◽  
Author(s):  
V. A. Malyshev
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Queueing Networks ◽  
Random Perturbation ◽  
Point Of View ◽  
Random Perturbations ◽  
Continuous Invariant Measure ◽  
New Approach ◽  
Lyapounov Exponent

A new approach to the problem of classification of (deflected) random walks in or Markovian models for queueing networks with identical customers is introduced. It is based on the analysis of the intrinsic dynamical system associated with the random walk. Earlier results for small dimensions are presented from this novel point of view. We give proofs of new results for higher dimensions related to the existence of a continuous invariant measure for the underlying dynamical system. Two constants are shown to be important: the free energy M < 0 corresponds to ergodicity, the Lyapounov exponent L < 0 defines recurrence. General conjectures, examples, unsolved problems and surprising connections with ergodic theory, classical dynamical systems and their random perturbations are largely presented. A useful notion naturally arises, the so-called scaled random perturbation of a dynamical system.

scholarly journals Networks and dynamical systems

1993 ◽  
Vol 25 (01) ◽  
pp. 140-175 ◽  
Author(s):  
V. A. Malyshev
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Queueing Networks ◽  
Random Perturbation ◽  
Point Of View ◽  
Random Perturbations ◽  
Continuous Invariant Measure ◽  
New Approach ◽  
Lyapounov Exponent

A new approach to the problem of classification of (deflected) random walks inor Markovian models for queueing networks with identical customers is introduced. It is based on the analysis of the intrinsic dynamical system associated with the random walk. Earlier results for small dimensions are presented from this novel point of view. We give proofs of new results for higher dimensions related to the existence of a continuous invariant measure for the underlying dynamical system. Two constants are shown to be important: the free energyM&lt; 0 corresponds to ergodicity, the Lyapounov exponentL&lt; 0 defines recurrence. General conjectures, examples, unsolved problems and surprising connections with ergodic theory, classical dynamical systems and their random perturbations are largely presented. A useful notion naturally arises, the so-called scaled random perturbation of a dynamical system.

scholarly journals On the Concept of Complexity of Random Dynamical Systems

1998 ◽  
Vol 12 (03) ◽  
pp. 225-243 ◽  
Author(s):  
V. Loreto ◽  
M. Serva ◽  
A. Vulpiani
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Random Perturbation ◽  
Point Of View ◽  
Random Dynamical Systems ◽  
Random Perturbations ◽  
Physical Point ◽  
Random Maps ◽  
Numerical Examples ◽  
Real Experimental Data

We show how to introduce a characterization the "complexity" of random dynamical systems. More precisely we propose a suitable indicator of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by these systems. This indicator of complexity, which can be extracted from real experimental data, turns out to be very natural in the context of information theory. For dynamical systems with random perturbations, it coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. In presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent λσ computed through the consideration of two nearby trajectories evolving under the same realization of the random perturbation. However, the former is much more relevant than the latter from a physical point of view as illustrated by some numerical examples of noisy and random maps.

Approximations of Dynamical Systems and Their Applications to Cryptography

2003 ◽  
Vol 13 (07) ◽  
pp. 1937-1948 ◽  
Author(s):  
J. M. Amigó ◽  
J. Szczepański
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Block Structure ◽  
Point Of View ◽  
Interval Exchange ◽  
Linear Cryptanalysis ◽  
New Approach ◽  
Interval Exchange Transformations ◽  
Measure Spaces ◽  
Dynamical Properties

During the last years a new approach to construct safe block and stream ciphers has been developed using the theory of dynamical systems. Since a block cryptosystem is generally, from the mathematical point of view, a family (parametrized by the keys) of permutations of n-bit numbers, one of the main problems of this approach is to adapt the dynamics defined by a map f to the block structure of the cryptosystem. In this paper we propose a method based on the approximation of f by periodic maps Tn (v.g. some interval exchange transformations). The approximation of automorphisms of measure spaces by periodic automorphisms was introduced by Halmos and Rohlin. One important aspect studied in our paper is the relation between the dynamical properties of the map f (say, ergodicity or mixing) and the immunity of the resulting cipher to cryptolinear attacks, which is currently one of the standard benchmarks for cryptosystems to be considered secure. Linear cryptanalysis, first proposed by M. Matsui, exploits some statistical inhomogeneities of expressions called linear approximations for a given cipher. Our paper quantifies immunity to cryptolinear attacks in terms of the approximation speed of the map f by the periodic Tn. We show that the most resistant block ciphers are expected when the approximated dynamical system is mixing.

Stability of Dynamical Systems With Multiple Nonlinearities

1987 ◽  
Vol 109 (4) ◽  
pp. 410-413 ◽  
Author(s):  
Norio Miyagi ◽  
Hayao Miyagi
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Stability Region ◽  
Direct Method ◽  
Essential Feature ◽  
Point Of View ◽  
Stability Of Dynamical Systems ◽  
The Stability

This note applies the direct method of Lyapunov to stability analysis of a dynamical system with multiple nonlinearities. The essential feature of the Lyapunov function used in this note is a non-Lure´ type Lyapunov function which surpasses the Lure´-type Lyapunov function from the point of view of the stability region guaranteed. A modified version of the multivariable Popov criterion is used to construct non-Lure´ type Lyapunov function, which allow for the dynamical sytems with multiple nonlinearities.

scholarly journals ON STOCHASTIC PERTURBATIONS OF DYNAMICAL SYSTEMS WITH FAST AND SLOW COMPONENTS

2001 ◽  
Vol 01 (02) ◽  
pp. 261-281 ◽  
Author(s):  
MARK FREIDLIN
Keyword(s):  
Dynamical Systems ◽  
Large Deviations ◽  
Stochastic Resonance ◽  
Slow Motion ◽  
Point Of View ◽  
Random Perturbations ◽  
Stochastic Perturbations ◽  
Large Deviations Theory ◽  
Stable Equilibria ◽  
Small Random Perturbations

Dynamical systems with fast and slow components are considered. We show that small random perturbations of the fast component can lead to essential changes in the limiting slow motion. For example, new stable equilibria or deterministic oscillations with amplitude and frequency of order 1 can be introduced by the perturbations. These are stochastic resonance type effects, and they are considered from the point of view of large deviations theory.

scholarly journals On a New Approach to Formation of a Strategy of Developing Small Businesses

2019 ◽  
Vol 29 (3) ◽  
pp. 484-490
Author(s):  
Boris Turenko ◽  
Tatyana Turenko
Keyword(s):  
Development Strategy ◽  
Small Businesses ◽  
Point Of View ◽  
Small Enterprises ◽  
Financial Strategy ◽  
Strategy Formation ◽  
New Approach ◽  
Points Of View ◽  
Further Development

An enterprise operating in terms of market economy must always set itself the task of further development and improvement in order to be successful and competitive. To solve this problem, it is necessary to produce a development strategy. This problem is all the more relevant for small enterprises, which find it more difficult than large enterprises to survive in the competition. Therefore, small businesses face the task of developing and implementing such a strategy for their development, which allows them to clearly define their goals and outline the ways to achieve them. The article considers various points of view on the concept of «strategy», the classification of strategies that exist in the scientific literature, including those applied to small businesses, brings forth the authors' point of view on this concept. It pays a particular attention to the proposed approach to the process of strategy formation and the mechanism of its implementation in relation to small businesses. It makes a conclusion that on the basis of the systematic approach the strategy under development should be comprehensive and include such elements as marketing, product, production, personnel, management, financial strategy and risk strategies. It brings forth a description of the content of each element. It examines an algorithm for developing an integrated strategy for development of small businesses, as well as a mechanism for implementing the strategy.

METASTABILITY FOR RANDOM PERTURBATIONS OF NEARLY-HAMILTONIAN SYSTEMS

2008 ◽  
Vol 08 (01) ◽  
pp. 1-21 ◽  
Author(s):  
AVANTI ATHREYA ◽  
MARK FREIDLIN
Keyword(s):  
Dynamical System ◽  
Stable Equilibrium ◽  
Random Perturbation ◽  
Equilibrium Points ◽  
Stochastic Perturbation ◽  
Averaging Principle ◽  
Random Perturbations ◽  
Small Random Perturbation ◽  
Finite Set ◽  
Single State

We characterize the phenomenon of metastability for a small random perturbation of a nearly-Hamiltonian dynamical system. We use the averaging principle and the theory of large deviations to prove that the metastable "state" is, in general, not a single state but rather a nondegenerate probability measure across the stable equilibrium points of the unperturbed Hamiltonian system. The set of all possible "metastable distributions" is a finite set that is independent of the stochastic perturbation.

scholarly journals Classification Of Road Accidents From The Perspective Of Vehicle Safety Systems

2015 ◽  
Vol 13 (2) ◽  
pp. 1-9
Author(s):  
Václav Jirovský
Keyword(s):  
Road Accident ◽  
Point Of View ◽  
Automated System ◽  
Collision Probability ◽  
New Approach ◽  
Time To Collision ◽  
Reaction Space ◽  
Safety Systems ◽  
Level Of Importance

Abstract Modern road accident investigation and database structures are focused on accident analysis and classification from the point of view of the accident itself. The presented article offers a new approach, which will describe the accident from the point of view of integrated safety vehicle systems. Seven main categories have been defined to specify the level of importance of automated system intervention. One of the proposed categories is a new approach to defining the collision probability of an ego-vehicle with another object. This approach focuses on determining a 2-D reaction space, which describes all possible positions of the vehicle or other moving object in the specified amount of time in the future. This is to be used for defining the probability of the vehicles interacting - when the intersection of two reaction spaces exists, an action has to be taken on the side of ego-vehicle. The currently used 1-D quantity of TTC (time-to-collision) can be superseded by the new reaction space variable. Such new quantity, whose basic idea is described in the article, enables the option of counting not only with necessary braking time, but mitigation by changing direction is then easily feasible. Finally, transparent classification measures of a probable accident are proposed. Their application is highly effective not only during basic accident comparison, but also for an on-board safety system.

scholarly journals Slow Invariant Manifolds of Slow–Fast Dynamical Systems

2021 ◽  
Vol 31 (07) ◽  
pp. 2150112
Author(s):  
Jean-Marc Ginoux
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Singular Perturbation ◽  
Invariant Manifolds ◽  
The Other ◽  
Singularly Perturbed ◽  
Van Der Pol ◽  
Flow Curvature ◽  
The One

Slow–fast dynamical systems, i.e. singularly or nonsingularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating their equations. This paper aims, on the one hand, to propose a classification of the most important of them into two great categories: singular perturbation-based methods and curvature-based methods, and on the other hand, to prove the equivalence between any methods belonging to the same category and between the two categories. Then, a deep analysis and comparison between each of these methods enable to state the efficiency of the Flow Curvature Method which is exemplified with paradigmatic Van der Pol singularly perturbed dynamical system and Lorenz slow–fast dynamical system.

scholarly journals A symbolic projection of Langton's Ant

2003 ◽  
Vol DMTCS Proceedings vol. AB,... (Proceedings) ◽  
Author(s):  
Anahi Gajardo
Keyword(s):  
Point Of View ◽  
Algebraic Characterization ◽  
Transition Rule ◽  
Regular Graphs ◽  
New Approach ◽  
The Family ◽  
International Audience ◽  
Topological Dynamical Systems

International audience The Langton's ant is studied from the point of view of topological dynamical systems. A new approach which associate a subshift to the system is proposed.The transition rule is generalized to the family of bi-regular graphs $\Gamma(k,d)$ and the dependence of the dynamical system on $k$ and $d$ is analyzed. A classification of the $\Gamma (k,d)$ graphs based on the dynamical properties of the subshift is established. Also a hierarchy is defined on the graphs through the subset relation of the respective subshifts. The analysis are worked out by establishing an algebraic characterization of the forbidden words of the subshift.

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