Simple dynamical systems

<div class="page" title="Page 1"><div class="layoutArea"><div class="column">In this paper, we study the class of simple systems on R induced by homeomorphisms having finitely many non-ordinary points. We characterize the family of homeomorphisms on R having finitely many non-ordinary points upto (order) conjugacy. For x,y ∈ R, we say x ∼ y on a dynamical system (R,f) if x and y have same dynamical properties, which is an equivalence relation. Said precisely, x ∼ y if there exists an increasing homeomorphism h : R → R such that h ◦ f = f ◦ h and h(x) = y. An element x ∈ R is ordinary in (R,f) if its equivalence class [x] is a neighbourhood of it. </div></div></div>

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Dynamics of Neural Networks as Nonlinear Systems with Several Equilibria

Advancing Artificial Intelligence through Biological Process Applications ◽

10.4018/978-1-59904-996-0.ch018 ◽

2011 ◽

pp. 331-357 ◽

Cited By ~ 6

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.

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STATISTICAL LAWS FOR COMPUTATIONAL COLLAPSE OF DISCRETIZED CHAOTIC MAPPINGS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496001533 ◽

1996 ◽

Vol 06 (12a) ◽

pp. 2389-2399 ◽

Cited By ~ 8

Author(s):

PHIL DIAMOND ◽

ALEXEI POKROVSKII

Keyword(s):

Dynamical System ◽

Fixed Point ◽

Dynamical Systems ◽

Close Agreement ◽

Computer Experiments ◽

Continuum State ◽

Chaotic Dynamical Systems ◽

The Family ◽

Random Mappings ◽

Statistical Laws

When a dynamical system is realized on a computer, the computation is of a discretization, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretizations often have collapsing effects to a fixed point or to short cycles. Statistical properties of this phenomenon can be modeled by random mappings with an absorbing center. The model gives results which are very much in line with computational experiments and there appears to be a type of universality summarised by an Arcsine Law. The effects are discussed with special reference to the family of mappings fl(x)=1−|1−2x|l,x∈[0, 1], 1<l≤2. Computer experiments show close agreement with predictions of the model.

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Repelling invariant curves in planar discrete dynamical systems

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700016579 ◽

1994 ◽

Vol 49 (3) ◽

pp. 469-481 ◽

Cited By ~ 1

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.

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Approximations of Dynamical Systems and Their Applications to Cryptography

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403007771 ◽

2003 ◽

Vol 13 (07) ◽

pp. 1937-1948 ◽

Cited By ~ 7

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.

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

SERIES III - MATEMATICS, INFORMATICS, PHYSICS ◽

10.31926/but.mif.2020.13.62.2.3 ◽

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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Dynamical properties of spatial discretizations of a generic homeomorphism

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.108 ◽

2014 ◽

Vol 35 (5) ◽

pp. 1474-1523 ◽

Cited By ~ 4

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.

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Integrable two-dimensional dynamical systems and the characteristic Lie algebras

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2007.1.023 ◽

2007 ◽

Vol 5 ◽

pp. 195-200

Author(s):

A.V. Zhiber ◽

O.S. Kostrigina

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Lie Algebra ◽

Lie Algebras ◽

Second Order ◽

System Of Equations ◽

Two Dimensional ◽

Finite Dimensional ◽

Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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Potential Well in Poincaré Recurrence

Entropy ◽

10.3390/e23030379 ◽

2021 ◽

Vol 23 (3) ◽

pp. 379

Author(s):

Miguel Abadi ◽

Vitor Amorim ◽

Sandro Gallo

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Stochastic Processes ◽

Recurrence Time ◽

Potential Well ◽

Exponential Approximation ◽

Mixing Processes ◽

System Perspective ◽

Return Times ◽

Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

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A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

Modern Physics Letters B ◽

10.1142/s0217984989001813 ◽

1989 ◽

Vol 03 (15) ◽

pp. 1185-1188 ◽

Cited By ~ 1

Author(s):

J. SEIMENIS

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Infinite Number ◽

Chaotic Behavior ◽

Equations Of Motion ◽

Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

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The Dynamics of Musical Participation

Musicae Scientiae ◽

10.1177/1029864920988319 ◽

2021 ◽

pp. 102986492098831

Author(s):

Andrea Schiavio ◽

Pieter-Jan Maes ◽

Dylan van der Schyff

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Systems Theory ◽

Dynamical Systems Theory ◽

Coordination Dynamics ◽

Musical Experience ◽

The Core ◽

Interactive Dynamics ◽

Interdisciplinary Scholarship ◽

Core Features

In this paper we argue that our comprehension of musical participation—the complex network of interactive dynamics involved in collaborative musical experience—can benefit from an analysis inspired by the existing frameworks of dynamical systems theory and coordination dynamics. These approaches can offer novel theoretical tools to help music researchers describe a number of central aspects of joint musical experience in greater detail, such as prediction, adaptivity, social cohesion, reciprocity, and reward. While most musicians involved in collective forms of musicking already have some familiarity with these terms and their associated experiences, we currently lack an analytical vocabulary to approach them in a more targeted way. To fill this gap, we adopt insights from these frameworks to suggest that musical participation may be advantageously characterized as an open, non-equilibrium, dynamical system. In particular, we suggest that research informed by dynamical systems theory might stimulate new interdisciplinary scholarship at the crossroads of musicology, psychology, philosophy, and cognitive (neuro)science, pointing toward new understandings of the core features of musical participation.

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