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 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.

Dynamics of Neural Networks as Nonlinear Systems with Several Equilibria

2011 ◽  
pp. 331-357 ◽  
Author(s):  
Daniela Danciu
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Time Delay ◽  
Dynamical Systems ◽  
Qualitative Properties ◽  
The Neural Network ◽  
The Neural Networks ◽  
Global Asymptotics ◽  
Dynamical Properties

Neural networks—both natural and artificial, are characterized by two kinds of dynamics. The first one is concerned with what we would call “learning dynamics”. The second one is the intrinsic dynamics of the neural network viewed as a dynamical system after the weights have been established via learning. The chapter deals with the second kind of dynamics. More precisely, since the emergent computational capabilities of a recurrent neural network can be achieved provided it has suitable dynamical properties when viewed as a system with several equilibria, the chapter deals with those qualitative properties connected to the achievement of such dynamical properties as global asymptotics and gradient-like behavior. In the case of the neural networks with delays, these aspects are reformulated in accordance with the state of the art of the theory of time delay 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 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.

Periodic Motions in a Discontinuous Dynamical System With Two Circular Boundaries

2019 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Periodic Motions ◽  
Discontinuous Dynamical System ◽  
Mapping Structures

Abstract In this paper, periodic motions in a discontinuous dynamical systems are studied. The discontinuous dynamical system has three domains partitioned through two circular boundaries. On the three domains, there are three distinct dynamical systems. From the G-functions, the switchability conditions of a flow from one domain to anther domain at the boundary are developed. The flow mappings from a boundary to a bounbary are developed for each domain and boundary. From the mapping structures, periodic motions in the discontinuous dynamical system are predicted. Numerical simulations of periodic motions and motion switchability at boundaries are presented in the discontinuous dynamical system.

scholarly journals Repelling invariant curves in planar discrete dynamical systems

1994 ◽  
Vol 49 (3) ◽  
pp. 469-481 ◽  
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Continuous Dependence ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Invariant Curves ◽  
Dynamical Properties ◽  
Attracting Fixed Point

Constructive, simple proofs for the existence, regularity, continuous dependence and dynamical properties of a repelling invariant curve for a discrete dynamical system of the plane with an attracting fixed point with real eigenvalues are given. These proofs can be used to generate a numerical algorithm to find these curves and to compute explicitly the dependence of the curve with respect to the system.

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.

scholarly journals A General Metric for the Similarity of Both Stochastic and Deterministic System Dynamics

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1191
Author(s):  
Colin Shea-Blymyer ◽  
Subhradeep Roy ◽  
Benjamin Jantzen
Keyword(s):  
Dynamical Systems ◽  
Structural Changes ◽  
Time Series Data ◽  
Partial Information ◽  
Series Data ◽  
Deterministic System ◽  
Deterministic Systems ◽  
Dynamical Features ◽  
Dynamical Properties ◽  
Well Posed

Many problems in the study of dynamical systems—including identification of effective order, detection of nonlinearity or chaos, and change detection—can be reframed in terms of assessing the similarity between dynamical systems or between a given dynamical system and a reference. We introduce a general metric of dynamical similarity that is well posed for both stochastic and deterministic systems and is informative of the aforementioned dynamical features even when only partial information about the system is available. We describe methods for estimating this metric in a range of scenarios that differ in respect to contol over the systems under study, the deterministic or stochastic nature of the underlying dynamics, and whether or not a fully informative set of variables is available. Through numerical simulation, we demonstrate the sensitivity of the proposed metric to a range of dynamical properties, its utility in mapping the dynamical properties of parameter space for a given model, and its power for detecting structural changes through time series data.

scholarly journals On forward iterated Hausdorffness and development of embryo from zygote in bitopological dynamical systems

2021 ◽  
Vol 13(62) (2) ◽  
pp. 399-410
Author(s):  
Santanu Acharjee ◽  
Kabindra Goswami ◽  
Hemanta Kumar Sarmah
Keyword(s):  
Dynamical System ◽  
Cell Division ◽  
Dynamical Systems ◽  
Topological Space ◽  
Development Process ◽  
Hausdorff Space ◽  
Topological Dynamical System ◽  
Relative Topology ◽  
Dynamical Properties

Topological dynamical system is an area of dynamical system to investigate dynamical properties in terms of a topological space. Nada and Zohny [Nada, S.I. and Zohny, H., An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202-204] applied topological dynamical system to explore the development process of an embryo from the zygote until birth and made three conjectures. In this paper, we disprove conjecture 3 of Nada and Zohny [Nada, S.I. and Zohny, H., An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202-204] by applying some of our mathematical results of bitopological dynamical system. Also, we introduce forward iterated Hausdorff space, backward iterated Hausdorff space, pairwise iterated Hausdor_ space and establish relations between them in bitopological dynamical system. We formulate the function that represents cell division (specially, mitosis) and using this function we show that in the development process of a human baby from the zygote until its birth, there is a stage where the developing stage is forward iterated Hausdorff

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