Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

Abstract : The emerging discipline known as \"chaos theory\" is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for \"chaos\" as it applies to general dynamical systems. Various approaches range from topological methods of a qualitative description; to physical notions of randomness, information, and entropy in ergodic theory; to the development of computational definitions and algorithms designed to obtain quantitative information. This thesis develops some of the current definitions and discusses several quantitative measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems, it serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.

Introduction of chaotic dynamical systems

In this paper, a digital image encryption algorithm is proposed based on the generalized model of the chaos game. The chaos game is a well-known fractal, which acts as a pseudo-random number generator (PRNG) in the proposed encryption algorithm. The foundation of the chaos game is based on basic points and its distance ratio that determine the basis of how they distribute random values in 2D or 3D space. These basic points are entered by the user interface and are the result of an encrypted image with a fractal structure. The use of the bifurcation diagram and Lyapunov exponent analysis showed that the proposed chaos game has the dynamical behavior, and fully chaotic characteristic, and can be used as a secure PRNG in cryptography systems. In the proposed method, the region of interest is determined by a number of Bases, and the fractal mechanism of chaos game for the encryption process is performed on the image. This process is very sensitive to any changes in keys and refers to confusion. The evaluation results of security and performance analysis on standard images confirm the efficiency of the proposed method and demonstrate that the proposed method is robust against attacks.

Deterministic chaos game: A new fractal based pseudo-random number generator and its cryptographic application

Hierarchy of one-dimensional ergodic chaotic maps with Tsallis type of q -deformation are studied. We find that in the chaotic region, these maps with q -deformation are ergodic as the Birkhoff ergodic theorem predicts. q -deformed maps are defined as ratios of polynomials of degree N . Hence, by using the Stieltjes transform approach (STA), invariant measure is proposed. In addition, considering Sinai-Ruelle-Bowen (SRB) measure, Kolmogorov-Sinai (KS) entropy for q -deformed maps is calculated analytically. The new q -deformed scheme have ability to keep previous significant properties such as ergodicity, sensitivity to initial condition. By adding q -parameter to the hierarchy in order increase the randomness and one-way computation, we present a new scheme for watermarking. The introduced algorithm tries to improve the problem of failure of encryption such as small key space, encryption speed and level of security. To illustrate the effectiveness of the proposed scheme, some security analyses are presented. By considering the obtained results, it can be concluded that, this scheme have a high potential to be adopted for watermarking. It can be concluded that, the proposed novel watermarking scheme for image authentication can be applied for practical applications.

/pdf/watermarking-based-on-discrete-wavelet-transform-and-q-4hn2jtorsl.pdf

Watermarking based on discrete wavelet transform and q-deformed chaotic map

A robust digital video watermarking based on CT-SVD domain and chaotic DNA sequences for copyright protection

In this letter a new watermarking scheme for Gray scale image is proposed based on a family of the chaotic maps and Singular Value Decomposition. Jacobian elliptic map is used to encrypt the watermark logo to improve the security of watermarked image. Quantum map is also used to determine the location of image’s block for the watermark embedding. To test the robustness and effectiveness of our proposed method, several attacks are applied to the watermarked image and the best results have been reported. The purpose of this algorithm is to improve the shortcoming of watermarking such as small key space and low security. The experimental results demonstrate that the key space is large enough to resist the attack and the distribution of grey values of the encrypted image has a random-like behavior, which makes it a potential candidate for encryption of multimedia data such as images, audios and even videos.

A SVD-Chaos Digital Image Watermarking Scheme Based on Multiple Chaotic System

In this letter a new watermarking scheme for Gray scale image is proposed based on a family of the chaotic maps and Singular Value Decomposition. Jacobian elliptic map is used to encrypt the watermark logo to improve the security of watermarked image. Quantum map is also used to determine the location of image's block for the watermark embedding. To test the robustness and effectiveness of our proposed method, several attacks are applied to the watermarked image and the best results have been reported. The purpose of this algorithm is to improve the shortcoming of watermarking such as small key space and low security. The experimental results demonstrate that the key space is large enough to resist the attack and the distribution of grey values of the encrypted image has a random-like behavior, which makes it a potential candidate for encryption of multimedia data such as images, audios and even videos.

/pdf/a-chaotic-blind-digital-image-watermarking-based-on-singular-4o1rue93di.pdf

A Chaotic Blind Digital Image Watermarking Based On Singular Value Decomposition in Spatial Domain

/pdf/a-svd-chaos-digital-image-watermarking-scheme-based-on-5bhgf7ecqy.pdf

Niaz Khorrami

Papers

A SVD-Chaos Digital Image Watermarking Scheme Based on Multiple Chaotic System

A Chaotic Blind Digital Image Watermarking Based On Singular Value Decomposition in Spatial Domain

A SVD-Chaos Digital Image Watermarking Scheme Based on Multiple Chaotic System