Numerics and Fractals

Local iterated function systems are an important generalisation of the standard (global) iterated function systems (IFSs). For a particular class of mappings, their fixed points are the graphs of local fractal functions and these functions themselves are known to be the fixed points of an associated Read-Bajactarevi\'c operator. This paper establishes existence and properties of local fractal functions and discusses how they are computed. In particular, it is shown that piecewise polynomials are a special case of local fractal functions. Finally, we develop a method to compute the components of a local IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note: substantial text overlap with arXiv:1309.0243. text overlap with arXiv:1309.024

Herding as a Learning System with Edge-of-Chaos Dynamics

Herding defines a deterministic dynamical system at the edge of chaos. It generates a sequence of model states and parameters by alternating parameter perturbations with state maximizations, where the sequence of states can be interpreted as "samples" from an associated MRF model. Herding differs from maximum likelihood estimation in that the sequence of parameters does not converge to a fixed point and differs from an MCMC posterior sampling approach in that the sequence of states is generated deterministically. Herding may be interpreted as a"perturb and map" method where the parameter perturbations are generated using a deterministic nonlinear dynamical system rather than randomly from a Gumbel distribution. This chapter studies the distinct statistical characteristics of the herding algorithm and shows that the fast convergence rate of the controlled moments may be attributed to edge of chaos dynamics. The herding algorithm can also be generalized to models with latent variables and to a discriminative learning setting. The perceptron cycling theorem ensures that the fast moment matching property is preserved in the more general framework

Graph dynamics : learning and representation

Thesis (S.M.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 2006.Includes bibliographical references (p. 58-60).Graphs are often used in artificial intelligence as means for symbolic knowledge representation. A graph is nothing more than a collection of symbols connected to each other in some fashion. For example, in computer vision a graph with five nodes and some edges can represent a table - where nodes correspond to particular shape descriptors for legs and a top, and edges to particular spatial relations. As a framework for representation, graphs invite us to simplify and view the world as objects of pure structure whose properties are fixed in time, while the phenomena they are supposed to model are actually often changing. A node alone cannot represent a table leg, for example, because a table leg is not one structure (it can have many different shapes, colors, or it can be seen in many different settings, lighting conditions, etc.) Theories of knowledge representation have in general concentrated on the stability of symbols - on the fact that people often use properties that remain unchanged across different contexts to represent an object (in vision, these properties are called invariants). However, on closer inspection, objects are variable as well as stable. How are we to understand such problems? How is that assembling a large collection of changing components into a system results in something that is an altogether stable collection of parts?(cont.) The work here presents one approach that we came to encompass by the phrase "graph dynamics". Roughly speaking, dynamical systems are systems with states that evolve over time according to some lawful "motion". In graph dynamics, states are graphical structures, corresponding to different hypothesis for representation, and motion is the correction or repair of an antecedent structure. The adapted structure is an end product on a path of test and repair. In this way, a graph is not an exact record of the environment but a malleable construct that is gradually tightened to fit the form it is to reproduce. In particular, we explore the concept of attractors for the graph dynamical system. In dynamical systems theory, attractor states are states into which the system settles with the passage of time, and in graph dynamics they correspond to graphical states with many repairs (states that can cope with many different contingencies). In parallel with introducing the basic mathematical framework for graph dynamics, we define a game for its control, its attractor states and a method to find the attractors. From these insights, we work out two new algorithms, one for Bayesian network discovery and one for active learning, which in combination we use to undertake the object recognition problem in computer vision. To conclude, we report competitive results in standard and custom-made object recognition datasets.by Andre Figueiredo Ribeiro.S.M

Deterministic Chaos in Digital Cryptography

This thesis studies the application of deterministic chaos to digital cryptography. Cryptographic systems such as pseudo-random generators (PRNG), block ciphers and hash functions are regarded as a dynamic system (X, j), where X is a state space (Le. message space) and f : X -+ X is an iterated function. In both chaos theory and cryptography, the object of study is a dynamic system that performs an iterative nonlinear transformation of information in an apparently unpredictable but deterministic manner. In terms of chaos theory, the sensitivity to the initial conditions together with the mixing property ensures cryptographic confusion (statistical independence) and diffusion (uniform propagation of plaintext and key randomness into cihertext). This synergetic relationship between the properties of chaotic and cryptographic systems is considered at both the theoretical and practical levels: The theoretical background upon which this relationship is based, includes discussions on chaos, ergodicity, complexity, randomness, unpredictability and entropy. Two approaches to the finite-state implementation of chaotic systems (Le. pseudo-chaos) are considered: (i) floating-point approximation of continuous-state chaos; (ii) binary pseudo-chaos. An overview is given of chaotic systems underpinning cryptographic algorithms along with their strengths and weaknesses. Though all conventional cryposystems are considered binary pseudo-chaos, neither chaos, nor pseudo-chaos are sufficient to guarantee cryptographic strength and security. A dynamic system is said to have an analytical solution Xn = (xo) if any trajectory point Xn can be computed directly from the initial conditions Xo, without performing n iterations. A chaotic system with an analytical solution may have a unpredictable multi-valued map Xn+l = f(xn). Their floating-point approximation is studied in the context of pseudo-random generators. A cryptographic software system E-Larm ™ implementing a multistream pseudo-chaotic generator is described. Several pseudo-chaotic systems including the logistic map, sine map, tangent- and logarithm feedback maps, sawteeth and tent maps are evaluated by means of floating point computations. Two types of partitioning are used to extract pseudo-random from the floating-point state variable: (i) combining the last significant bits of the floating-point number (for nonlinear maps); and (ii) threshold partitioning (for piecewise linear maps). Multi-round iterations are produced to decrease the bit dependence and increase non-linearity. Relationships between pseudo-chaotic systems are introduced to avoid short cycles (each system influences periodically the states of other systems used in the encryption session). An evaluation of cryptographic properties of E-Larm is given using graphical plots such as state distributions, phase-space portraits, spectral density Fourier transform, approximated entropy (APEN), cycle length histogram, as well as a variety of statistical tests from the National Institute of Standards and Technology (NIST) suite. Though E-Larm passes all tests recommended by NIST, an approach based on the floating-point approximation of chaos is inefficient in terms of the quality/performance ratio (compared with existing PRNG algorithms). Also no solution is known to control short cycles. In conclusion, the role of chaos theory in cryptography is identified; disadvantages of floating-point pseudo-chaos are emphasized although binary pseudo-chaos is considered useful for cryptographic applications.Durand Technology Limite

Distributed video through telecommunication networks using fractal image compression techniques

The research presented in this thesis investigates the use of fractal compression techniques for a real time video distribution system. The motivation for this work was that the method has some useful properties which satisfy many requirements for video compression. In addition, as a novel technique, the fractal compression method has a great potential. In this thesis, we initially develop an understanding of the state of the art in image and video compression and describe the mathematical concepts and basic terminology of the fractal compression algorithm. Several schemes which aim to the improve of the algorithm, for still images are then examined. Amongst these, two novel contributions are described. The first is the partitioning of the image into sections which resulted insignificant reduction of the compression time. In the second, the use of the median metric as alternative to the RMS was considered but was not finally adopted, since the RMS proved to be a more efficient measure. The extension of the fractal compression algorithm from still images to image sequences is then examined and three different schemes to reduce the temporal redundancy of the video compression algorithm are described. The reduction in the execution time of the compression algorithm that can be obtained by the techniques described is significant although real time execution has not yet been achieved. Finally, the basic concepts of distributed programming and networks, as basic elements of a video distribution system, are presented and the hardware and software components of a fractal video distribution system are described. The implementation of the fractal compression algorithm on a TMS320C40 is also considered for speed benefits and it is found that a relatively large number of processors are needed for real time execution

A reduced-order model for segregated fluid-structure interaction solvers based on an ALE approach

This article presents a Galerkin projection model order reduction approach for fluid-structure interaction (FSI) problems in the Finite Volume context. The reduced-order model (ROM) is based on proper orthogonal decomposition (POD), where a reduced basis is formed using energy-dominant POD modes. The reduced basis also consists of characteristics of the POD time modes derived from the POD time modes coefficients. In addition, the solution state vector comprises the mesh deformation, considering the structural motion in FSI. The results are obtained by applying the proposed method to time-dependent problems governed by the 2D incompressible Navier-Stokes equations. The main objective of this work is to introduce a hybrid technique mixing up the classical Galerkin-projection approach with a data-driven method to obtain a versatile and accurate algorithm for resolving FSI problems with moving meshes. The effectiveness of this approach is demonstrated in the case study of vortex-induced vibrations (VIV) of a cylinder at Reynolds number Re = 200. The results show the stability and accuracy of the proposed method with respect to the high-dimensional model by capturing transient flow fields and, more importantly, the forces acting on the moving objects

A complex dynamical systems approach to the development of feeding problems in early childhood

Though it is commonly agreed upon that the development of feeding problems in early childhood is a complex process, much of the research on these problems has a component-oriented focus, and very little attention is paid to the mechanisms that lead to these kinds of problems in individual children. The aim of this theoretical paper is to interpret the development of feeding problems in early childhood from a complex dynamical systems viewpoint. In addition to its focus on self-organization and nonlinearity, this approach defines several central properties of development: soft-assembly, embodiment, iterativity, the emergence of higher-order properties, and intra-individual variability. In this paper, I argue that each of these properties is highly relevant for understanding feeding problems and discuss the implications of this for both clinical practice and research purposes

Research on digital image watermark encryption based on hyperchaos

The digital watermarking technique embeds meaningful information into one or more watermark images hidden in one image, in which it is known as a secret carrier. It is difficult for a hacker to extract or remove any hidden watermark from an image, and especially to crack so called digital watermark. The combination of digital watermarking technique and traditional image encryption technique is able to greatly improve anti-hacking capability, which suggests it is a good method for keeping the integrity of the original image. The research works contained in this thesis include: (1)A literature review the hyperchaotic watermarking technique is relatively more advantageous, and becomes the main subject in this programme. (2)The theoretical foundation of watermarking technologies, including the human visual system (HVS), the colour space transform, discrete wavelet transform (DWT), the main watermark embedding algorithms, and the mainstream methods for improving watermark robustness and for evaluating watermark embedding performance. (3) The devised hyperchaotic scrambling technique it has been applied to colour image watermark that helps to improve the image encryption and anti-cracking capabilities. The experiments in this research prove the robustness and some other advantages of the invented technique. This thesis focuses on combining the chaotic scrambling and wavelet watermark embedding to achieve a hyperchaotic digital watermark to encrypt digital products, with the human visual system (HVS) and other factors taken into account. This research is of significant importance and has industrial application value