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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
2D mixed pseudo-random coupling PS map lattice and its application in S-box generation
In this paper, firstly, we investigate a new 1D PWLCM-Sin (PS) map which derived from PWLCM and Sin map by modulo operation. Due to the stronger parameter space, bigger Lyapunov exponents and better ergodicity than simple 1D map, the PS map is more suitable for local map of spatiotemporal dynamics. Secondly, with the novel 2D pseudo-random mixed coupling method we present a spatiotemporal chaos which used PS map as local map
f
(
x
). This spatiotemporal chaos named 2D Mixed pseudo-random Coupling PS Map Lattice (2DMCPML). The experimental results of bifurcation diagrams, Kolmogorov–Sinai entropy density and spatiotemporal chaotic diagrams showed that 2DMCPML has advantages of larger parameter space, more complex chaotic behavior and more ergodic output sequence than CML. Therefore, 2DMCPML is more suitable in cryptography than CML. Subsequently, we proposed a chaos-based random S-box design algorithm employed the spatial chaotic character of 2DMCPML to generate a large number of S-boxes. The cryptographic performance indicated that generated S-boxes can resist cryptanalysis attack well. Finally, four criteria bounds are set. The numbers of S-boxes satisfying these bounds generated by 2DMCPML and several 1D chaotic maps is calculated, respectively. The result showed that spatiotemporal chaos can generate more S-boxes with high cryptographic quality than low-dimensional chaos. This new discovery is significant to the development of some cryptographic researches such as dynamic S-box algorithm.
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
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
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
Chaos-based engineering applications with a 3D chaotic system without equilibrium points
There has recently been an increase in the number of new chaotic system designs and chaos-based engineering applications. In this study, since homoclinic and heteroclinic orbits did not exist and analyses like Shilnikov method could not be used, a 3D chaotic system without equilibrium points was included and thus different engineering applications especially for encryption studies were realized. The 3D chaotic system without equilibrium points represents a new different phenomenon and an almost unexplored field of research. First of all, chaotic system without equilibrium points was examined as the basis and electronic circuit application of the chaotic system was realized and oscilloscope outputs of phase portraits were obtained. Later, chaotic system without equilibrium points was modelled on Labview Field Programmable Gate Array (FPGA) and then FPGA chip statistics, phase portraits and oscilloscope outputs were derived. With another study, VHDL and RK-4 algorithm were used and a new FPGA-based chaotic oscillators design was achieved. Results of Labview-based design on FPGA- and VHDL-based design were compared. Results of chaotic oscillator units designed here were gained via Xilinx ISE Simulator. Finally, a new chaos-based RNG design was achieved and internationally accepted FIPS-140-1 and NIST-800-22 randomness tests were run. Furthermore, video encryption application and security analyses were carried out with the RNG designed here.
Journal Article