Definition Of Chaos Research Articles

Chaotic foliations with Ehresmann connection

We consider smooth codimension q foliations on n-dimensional manifolds where 0<q<n. We use Ehresmann connections as a technical tool to introduce the notion of sensitivity to initial conditions for foliations. We extend Devaney's definition of chaos for cascades to foliations with Ehresmann connection. Our main result states that sensitivity to initial conditions of a foliation with Ehresmann connection follows from topological transitivity and density of minimal sets of the foliation. Compactness both minimal sets and the ambient manifold is not assumed. The results are applied to complete Cartan foliations.

Some chaotic properties of anosov diffeomorphisms of n-dimention tours T^n

In this search, we review the relationship of Anosov diffeomorphism with hyperbolic dynamic systems, where we present some definitions and basic observations of it. We find that the map possesses topological transitive and sensitive dependence on initial conditions and other tools of chaos, and therefore the map has chaotic properties by using many definitions of chaos.

Replication in Energy Markets: Use and Misuse of Chaos Tools.

As pointed out by many researchers, replication plays a key role in the credibility of applied sciences and the confidence in all research findings. With regard, in particular, to energy finance and economics, replication papers are rare, probably because they are hampered by inaccessible data, but their aim is crucial. We consider two ways to avoid misleading results on the ostensible chaoticity of price series. The first one is represented by the proper mathematical definition of chaos and the related theoretical background, while the latter is represented by the hybrid approach that we propose here—i.e., consisting of considering the dynamical system underlying the price time series as a deterministic system with noise. We find that both chaotic and stochastic features coexist in the energy commodity markets, although the misuse of some tests in the established practice in the literature may say otherwise.

Study of Chaotic Behaviour with G-Bi-Shadowing Property

The aim of this work is to study the -bi-shadowing property of the metric -space. We generalized the results to the metric -space and studied the chaotic properties by introducing a new definition of chaos which we call --chaotic in a neighborhood of a set and comparing it with the definition of Li-York of chaos in -space. We will study the above definitions with -homoclinic orbit and -chain components.&#x0D; The main results that we obtained in this paper, for some conditions, is -the homoclinic orbit of, and is both --bi-shadowing and --periodic bi-shadowing on (when being an unordered set), then any action which satisfies some conditions is --chaotic on a neighborhood of. Second, for some conditions, if action is --expansive and both --bi-shadowing and --periodic bi-shadowing with respect to an action and is a -the homoclinic orbit of contained, then every action satisfying some conditions is --chaotic on a neighborhood of. Third, for some conditions, if be a -chain component of an action, and is both --bi-shadowing and --periodic bi-shadowing on a -chain recurrent set -, Then every action which satisfies some conditions is --chaotic in a neighborhood.

Robust Devaney chaos in the two-dimensional border-collision normal form.

The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on R2 can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove that throughout an open region of parameter space, this family has an attractor satisfying Devaney's definition of chaos. This strengthens the existing results on the robustness of chaos in piecewise-linear maps. We further show that the stable manifold of a saddle fixed point, despite being a one-dimensional object, densely fills an open region containing the attractor. Finally, we identify a heteroclinic bifurcation, not described previously, at which the attractor undergoes a crisis and may be destroyed.

Dynamic analysis and image encryption application of a sinusoidal-polynomial composite chaotic system

Due to complex properties of ergodicity, non-periodic ability and sensitivity to initial states, chaotic systems are widely used in cryptography. In this paper, we propose a sinusoidal-polynomial composite chaotic system (SPCCS), and prove that it satisfies Devaney’s definition of chaos: the sensitivity to initial conditions, topological transitivity and density of periodic points. The experimental results show that SPCCS has better unpredictability and more complex chaotic behavior than the classical chaotic maps. Furthermore, we provide a new image encryption algorithm combining pixel segmentation operation, block chaotic matrix confusing operation and pixel diffusion operation with the proposed SPCCS. The detailed simulation results verify superiority and effectiveness of the proposed image encryption algorithm.

A New Version of Distributional Chaos, Distributional Chaos in a Sequence, and Other Concepts of Chaos

In this paper, we investigate the relations between distributional chaos in a sequence and distributional chaos ([Formula: see text]-chaos, R–T chaos, DC3, respectively). Firstly, we prove a sufficient condition that the distributional chaos is equivalent to the distributional chaos in a sequence. Besides, we prove that distributional chaos in a sequence and [Formula: see text]-chaos (R–T chaos, DC3, respectively) do not imply each other. Finally, we give a new definition of chaos, named DC2[Formula: see text], which is similar to DC2, and show that DC2[Formula: see text] is an invariant of topological conjugacy and an iteration invariant (that is, for any integer [Formula: see text], [Formula: see text] is DC2[Formula: see text] if and only if [Formula: see text] is DC2[Formula: see text]).

Constructing Higher-Dimensional Digital Chaotic Systems via Loop-State Contraction Algorithm

This paper aims to refine and expand the theoretical and application framework of higher-dimensional digital chaotic system (HDDCS). Topological mixing for HDDCS is strictly proved theoretically at first. Topological mixing implies Devaney's definition of chaos in a compact space, but not vice versa. Therefore, the proof of topological mixing promotes the theoretical research of HDDCS. Then, a general design method for constructing HDDCS via loop-state contraction algorithm is given. The construction of the iterative function uncontrolled by random sequences (hereafter called iterative function) is the starting point of this research. On this basis, this paper put forward a general design method to solve the construction problem of HDDCS, and several examples illustrate the effectiveness and feasibility of this method. The adjacency matrix corresponding to the designed HDDCS is used to construct the chaotic Echo State Network (ESN) for predicting Mackey-Glass time series. Compared with other ESNs, the chaotic ESN has better prediction performance and is able to accurately predict a much longer period of time.

Genericity of chaos for colored graphs

Abstract To each colored graph one can associate its closure in the universal space of isomorphism classes of pointed colored graphs, and this subspace can be regarded as a generalized subshift. Based on this correspondence, we introduce two definitions for chaotic (colored) graphs, one of them analogous to Devaney’s. We show the equivalence of our two novel definitions of chaos, proving their topological genericity in various subsets of the universal space.

A constructive approach to robust chaos using invariant manifolds and expanding cones

Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049-3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vectors expand and so no invariant set can have only negative Lyapunov exponents. Under additional assumptions we characterise an attractor as the closure of the unstable manifold of a fixed point and prove that it satisfies Devaney's definition of chaos.

A novel method of constructing high-dimensional digital chaotic systems on finite-state automata**Project supported by the National Key R&D Program of China (Grant No. 2017YFB0802000) and the Cryptography Theoretical Research of National Cryptography Development Fund, China (Grant No. MMJJ20170109).

When chaotic systems are implemented on finite precision machines, it will lead to the problem of dynamical degradation. Aiming at this problem, most previous related works have been proposed to improve the dynamical degradation of low-dimensional chaotic maps. This paper presents a novel method to construct high-dimensional digital chaotic systems in the domain of finite computing precision. The model is proposed by coupling a high-dimensional digital system with a continuous chaotic system. A rigorous proof is given that the controlled digital system is chaotic in the sense of Devaney’s definition of chaos. Numerical experimental results for different high-dimensional digital systems indicate that the proposed method can overcome the degradation problem and construct high-dimensional digital chaos with complicated dynamical properties. Based on the construction method, a kind of pseudorandom number generator (PRNG) is also proposed as an application.

A Class of Quadratic Polynomial Chaotic Maps and Their Fixed Points Analysis.

When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly depends on the exhaustive search of systematic parameters or initial values, especially for a class of dynamical systems with hidden chaotic attractors. In this paper, a class of quadratic polynomial chaotic maps is studied, and a general method for constructing quadratic polynomial chaotic maps is proposed. The proposed polynomial chaotic maps satisfy the Li–Yorke definition of chaos. This method can accurately control the amplitude of chaotic time series. Through the existence and stability analysis of fixed points, we proved that such class quadratic polynomial maps cannot have hidden chaotic attractors.

Chaos in a linear wave equation

A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics. The system comprises a one-dimensional wave equation with gain that operates on a semi-infinite line. A boundary condition enforces that the waves remain finite. It is shown that the resulting solution set is dense with periodic orbits, contains transitive orbits, and exhibits extreme sensitivity to initial conditions. Definitions of chaos are considered in light of such linear chaos.

Dynamical behavior of the Family of cubic functions

May R.M. gave the example of the family of cubic maps of the interval . Rogers T. D. extends the analysis of May beyond that region . “&#x0D; In this paper we are trying to introduce comprehensive study of the cubic family which defined in the form:&#x0D; The fixed points of the family are determined and described according to the values of the parameter . Dynamical and chaotic behaviours of the family discussed according to different definitions of chaos and via conjugacy .

A New Pseudorandom Bit Generator Based on Mixing Three‐Dimensional Chen Chaotic System with a Chaotic Tactics

In this paper, a new chaotic system is proposed based on mixing three‐dimensional Chen chaotic system with a chaotic tactics. This new system is proved to be chaotic under Wiggins’ chaos definition and can generate chaotic sequences with high complexity. Furthermore, we propose a new pseudorandom bit generator (PRBG) based on this new system. A coding algorithm is used to make the sequences uniform. Both statistical and security tests are provided to show the generated sequences are with good randomness and high complexity to withstand attacks.

Distributional Chaos and Dendrites

Many definitions of chaos have appeared in the last decades and there is a concerned question if they are equivalent in some more specific spaces. Our focus will be on distributional chaos, first defined in 1994 and later subdivided into three major types (and even more subtypes). These versions of chaos are equivalent on a closed interval, but distinct in more complicated spaces. Since dendrites have much in common with the interval, we explore whether or not we can distinguish these kinds of chaos already on dendrites. At the end of the paper we will also briefly look at their correlation with other types of chaos.

Analyzing Devaney Chaos of a Sine–Cosine Compound Function System

The one-dimensional Sine map and Chebyshev map are classical chaotic maps, which have clear chaotic characteristics. In this paper, we establish a chaotic framework based on a Sine–Cosine compound function system by analyzing the existing one-dimensional Sine map and Chebyshev map. The sensitive dependence on initial conditions, topological transitivity and periodic-point density of this chaotic framework is proved, showing that the chaotic framework satisfies Devaney’s chaos definition. In order to illustrate the chaotic behavior of the chaotic framework, we propose three examples, called Cosine–Polynomial (C–P) map, Sine–Tangent (S–T) map and Sine–Exponent (S–E) map, respectively. Then, we evaluate the chaotic behavior with Sine map and Chebyshev map by analyzing bifurcation diagrams, Lyapunov exponents, correlation dimensions, Kolmogorov entropy and [Formula: see text] complexity. Experimental results show that the chaotic framework has better unpredictability and more complex chaotic behaviors than the classical Sine map and Chebyshev map. The results also verify the effectiveness of the theoretical analysis of the proposed chaotic framework.

A remark on banks theorem

ABSTRACTFor discrete autonomous dynamical systems, it was found that in the three conditions defining Devaney chaos, topological transitivity and dense periodic points together imply sensitive dependence on initial condition (Banks, Brooks, Cairns, Davis and Stacey, On Devaney's definition of chaos, The Amer. Math. Monthly 99 (1992), pp. 332–334). In this paper, we give the definition of finitely generated non-autonomous dynamical systems (NADS) and generalize the Banks et al. theorem to the finitely generated NADS.

Universal spectral correlations in the chaotic wave function and the development of quantum chaos

We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix $\rho_A$ of a spatial subsystem A. We find that the spectrum of $\rho_A$ is described by a so-called Wishart random matrix, which exhibits universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings. We use these universal spectral characteristics of $\rho_A$ as a definition of chaos in the wave function. A simple and precise characterization of such correlations is a segment of linear growth at sufficiently long times, recently called the "ramp", of the spectral form factor. Specifically, numerical results for the spectral form factor of $\rho_A$ of generic non-integrable systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit an universal ramp identical to that appearing for a random pure state, in which $\rho_A$ is the Wishart random matrix. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ramp set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of $\rho_A$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ramp in the spectral form factor of $\rho_A$, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized chaotic regime.

Chaos theory from the mathematical point of view

This work is intended as an attempt to survey existingde finitions of chaos for discrete dynamical systems. Discussion is restricted to the settingof topological dynamics, while the measure-theoretic (ergodic theory) and smooth (differentiable dynamical systems) aspects are omitted as exceedingt he scope of this paper. Chaos theory is understood here as a part of topological dynamics, so aforementioned definitions of chaos are just examples of particular dynamical system properties, and are considered inside the framework of the mathematical theory of discrete dynamical systems. It is not the purpose of this article to study chaos theory understood as a new kind of interdisciplinary branch of science devoted to nonlinear phenomena. As for prerequisites, the reader is expected to possess some mathematical maturity, and to be familiar with basic topology of (compact) metric spaces. No preliminary knowledge of the dynamical systems theory is required, however some is recommended. The first two section are devoted to general discussion of the term „chaos” and contains authors opinion on this subject. To facilitate access to the rest of the article some relevant material from the dynamical system theory is briefly repeated in the third section. The next section (Section 4) introduces the notion of topological transitivity along with some stronger variants, namely topological mixing and weak mixing. Section 5 gives a detailed account of the famous Sharkovskii’s Theorem in its full generality. This is required for characterization of chaotic interval maps. Sections 6-13 are devoted to various notions of chaos or related to chaos in dynamical systems. Each section contains an attempt to motivate the notion, historical background and formal definition followed with a review of known properties, relations between various notions of chaos, and some relevant open problems. Section 6 is devoted to a sensitivity to initial conditions – a notion which is accepted as a basic indicator of chaotic behavior. Section 7 introduces a definition of chaos accordingt o Auslander and Yorke. Section 8 examines the notion of Li-Yorke pair and Li-Yorke chaos. Section 9 deals with the definition of chaos introduced in Devaney’s book (Devaney chaos). Section 10 recalls some facts connected with symbolic dynamics, which provides a rich source of examples for various interestingb ehavior, and it is an indispensable tool for exploration of many systems. Section 11 describes the so-called “topological horseshoes”, which are generalizations of the famous example due to Smale. The existence of a horseshoe in a given dynamical system proves the existence of a subsystem with a dynamics similar to some symbolic dynamical system, hence with a very complicated behavior. Section 12 gives a brief exposition of the topological entropy and its relation to chaos. The review of various notions of chaos ends with section 13, containingd escription of distributional chaos.