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Fluke : chance, chaos, and why everything we do matters
A social scientist dispels people's tidy versions of reality and delves deeply into the theories of random chance and chaos to demonstrate that the world really works through random events that can alter the trajectory of our lives.
Book
Differential equations, dynamical systems, and an introduction to chaos
Hirsch, Devaney, and Smale's classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems. Classic text by three of the world's most prominent mathematicians Continues the tradition of expository excellenceContains updated material and expanded applications for use in applied studies
eBook
Chaos-Based Image Encryption: Review, Application, and Challenges
Chaos has been one of the most effective cryptographic sources since it was first used in image-encryption algorithms. This paper closely examines the development process of chaos-based image-encryption algorithms from various angles, including symmetric and asymmetric algorithms, block ciphers and stream ciphers, and integration with other technologies. The unique attributes of chaos, such as sensitivity to initial conditions, topological transitivity, and pseudo-randomness, are conducive to cross-referencing with other disciplines and improving image-encryption methods. Additionally, this paper covers practical application scenarios and current challenges of chaotic image encryption, thereby encouraging researchers to continue developing and complementing existing situations, and may also serve as a basis of future development prospects for chaos-based image encryption.
Journal Article
Recent Development of Chaos Theory in Topological Dynamics
We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.
Journal Article
Color image encryption using orthogonal Latin squares and a new 2D chaotic system
Recently, many image encryption schemes have been developed using Latin squares. When encrypting a color image, these algorithms treat the color image as three greyscale images and encrypt these greyscale images one by one using the Latin squares. Obviously, these algorithms do not sufficiently consider the inner connections between the color image and Latin square and thus result in many redundant operations and low efficiency. To address this issue, in this paper, we propose a new color image encryption algorithm (CIEA) that sufficiently considers the properties of the color image and Latin square. First, we propose a two-dimensional chaotic system called 2D-LSM to address the weaknesses of existing chaotic systems. Then, we design a new CIEA using orthogonal Latin squares and 2D-LSM. The proposed CIEA can make full use of the inherent connections of the orthogonal Latin squares and color image and executes the encryption process in the pixel level. Simulation and security analysis results show that the proposed CIEA has a high level of security and can outperform some representative image encryption algorithms.
Journal Article
Positive topological entropy implies chaos DC2
Using methods of entropy in ergodic theory, we prove that positive topological entropy implies chaos DC2. That is, if a system (X,T)(X,T) has positive topological entropy, then there exists an uncountable set EE such that for any two distinct points x,yx,y in EE, \\[ lim infn→∞1n∑i=1ndist(Tix,Tiy)=0 and lim supn→∞1n∑i=1ndist(Tix,Tiy)>0.\\liminf _{n\\to \\infty } \\frac 1n \\sum _{i=1}^n \\mathsf {dist}(T^ix,T^iy)=0 \\ \\ \\ \\text {and} \\ \\ \\limsup _{n\\to \\infty } \\frac 1n \\sum _{i=1}^n \\mathsf {dist}(T^ix,T^iy)>0 . \\]
Journal Article
Cancer and Chaos and the Complex Network Model of a Multicellular Organism
In the search of theoretical models describing cancer, one of promising directions is chaos. It is connected to ideas of “genome chaos” and “life on the edge of chaos”, but they profoundly differ in the meaning of the term “chaos”. To build any coherent models, notions used by both ideas should be firstly brought closer. The hypothesis “life on the edge of chaos” using deterministic chaos has been radically deepened developed in recent years by the discovery of half-chaos. This new view requires a deeper interpretation within the range of the cell and the organism. It has impacts on understanding “chaos” in the term “genome chaos”. This study intends to present such an interpretation on the basis of which such searches will be easier and closer to intuition. We interpret genome chaos as deterministic chaos in a large module of half-chaotic network modeling the cell. We observed such chaotic modules in simulations of evolution controlled by weaker variant of natural selection. We also discuss differences between free and somatic cells in modeling their disturbance using half-chaotic networks.
Journal Article
Is Weather Chaotic?
Over 50 years since Lorenz’s 1963 study and a follow-up presentation in 1972, the statement “weather is chaotic” has been well accepted. Such a view turns our attention from regularity associated with Laplace’s view of determinism to irregularity associated with chaos. In contrast to single-type chaotic solutions, recent studies using a generalized Lorenz model (GLM) have focused on the coexistence of chaotic and regular solutions that appear within the same model using the same modeling configurations but different initial conditions. The results, with attractor coexistence, suggest that the entirety of weather possesses a dual nature of chaos and order with distinct predictability. In this study, based on the GLM, we illustrate the following two mechanisms that may enable or modulate two kinds of attractor coexistence and, thus, contribute to distinct predictability: 1) the aggregated negative feedback of small-scale convective processes that can produce stable nontrivial equilibrium points and, thus, enable the appearance of stable steady-state solutions and their coexistence with chaotic or nonlinear oscillatory solutions, referred to as the first and second kinds of attractor coexistence; and 2) the modulation of large-scale time-varying forcing (heating) that can determine (or modulate) the alternative appearance of two kinds of attractor coexistence. Based on our results, we then discuss new opportunities and challenges in predictability research with the aim of improving predictions at extended-range time scales, as well as subseasonal to seasonal time scales.
Journal Article
Chaos Game Optimization: a novel metaheuristic algorithm
In this paper, a novel metaheuristic algorithm called Chaos Game Optimization (CGO) is developed for solving optimization problems. The main concept of the CGO algorithm is based on some principles of chaos theory in which the configuration of fractals by chaos game concept and the fractals self-similarity issues are in perspective. A total number of 239 mathematical functions which are categorized into four different groups are collected to evaluate the overall performance of the presented novel algorithm. In order to evaluate the results of the CGO algorithm, three comparative analysis with different characteristics are conducted. In the first step, six different metaheuristic algorithms are selected from the literature while the minimum, mean and standard deviation values alongside the number of function evaluations for the CGO and these algorithms are calculated and compared. A complete statistical analysis is also conducted in order to provide a valid judgment about the performance of the CGO algorithm. In the second one, the results of the CGO algorithm are compared to some of the recently developed fractal- and chaos-based algorithms. Finally, the performance of the CGO algorithm is compared to some state-of-the-art algorithms in dealing with the state-of-the-art mathematical functions and one of the recent competitions on single objective real-parameter numerical optimization named “CEC 2017” is considered as numerical examples for this purpose. In addition, a computational cost analysis is also conducted for the presented algorithm. The obtained results proved that the CGO is superior compared to the other metaheuristics in most of the cases.
Journal Article