Dynamics for chaos and fractals

Alejaily, Ejaily
In this thesis, we study how to construct and analyze dynamics for chaos and fractals. After the introductory chapter, we discuss in the second chapter the chaotic behavior of hydrosphere parameters and their influence on global weather and climate. For this purpose, we investigate the nature and source of unpredictability in the dynamics of sea surface temperature. The impact of sea surface temperature variability on the global climate is clear during some global climate patterns like the El Niño-Southern Oscillation. The interactions between these types of global climate patterns may transmit chaos. We discuss the unpredictability as a global phenomenon through extension of chaos horizontally and vertically by using theoretical as well as numerical analyses. In the third chapter, we introduce a technique concerning the construction of dynamics for fractals. Deterministic fractals like Julia sets are crucial for understanding the fractal phenomenon. These structures are deterministically arising from simple dynamics of iteration of analytic functions. On the basis of the dynamics, we develop a scheme to map fractals through iterations. This allows to involve fractals as states of dynamical systems as well as to introduce dynamics in fractals through differential and discrete equations. Creation of effective frameworks for applications of fractals and assisting to explore the relationship between fractals and chaos by the involvement of dynamics in fractals are important aspects in this direction.


Dynamics with Chaos and Fractals
Akhmet, Marat (Springer, London/Berlin , 2020-02-01)
Stands as the first book presenting theoretical background on the unpredictable point and mapping of fractals.Introduces the concepts of unpredictable functions, abstract self-similarity, and similarity map.Discusses unpredictable solutions of quasilinear ordinary and functional differential equations.Illustrates new ways to construct fractals based on the ideas of Fatou and Julia. Examines unpredictability in ocean dynamics and neural networks, chaos in hybrid systems on a time scale, and homoclinic and he...
Analysis of double-negative materials with surface integral equations and the multilevel fast multipole algorithm
Ergül, Özgür Salih (2011-08-13)
We present a fast and accurate analysis of double-negative materials (DNMs) with surface integral equations and the multilevel fast multipole algorithm (MLFMA). DNMs are commonly used as simplified models of metamaterials at resonance frequencies and are suitable to be formulated with surface integral equations. However, realistic metamaterials and their models are usually very large with respect to wavelength and their accurate solutions require fast algorithms, such as MLFMA. We consider iterative solutio...
Stable controller design for T-S fuzzy systems based on Lie algebras
Banks, SP; Gurkan, E; Erkmen, İsmet (Elsevier BV, 2005-12-01)
In this paper, we study the stability of fuzzy control systems of Takagi-Sugeno-(T-S) type based on the classical theory of Lie algebras. T-S fuzzy systems are used to model nonlinear systems as a set of rules with consequents of the type x(t) = A(l)x (t) + B(l)u (t). We conduct the stability analysis of such T-S fuzzy models using the Lie algebra LA generated by the A(l) matrices of these subsystems for each rule in the rule base. We first develop our approach of stability analysis for a commuting algebra ...
Numerical modelling and finite element analysis of stress wave propagation for ultrasonic pulse velocity testing of concrete
Yaman, İsmail Özgür; Aktan, Haluk (2006-12-01)
Stress wave propagation through concrete is simulated by finite element analysis. The concrete medium is modeled as a homogeneous material with smeared properties to investigate and establish the suitable finite element analysis method (explicit versus implicit) and analysis parameters (element size, and solution time increment) also suitable for rigorous investigation. In the next step, finite element analysis model of the medium is developed using a digital image processing technique, which distinguishes ...
Rigorous Analysis of Double-Negative Materials with the Multilevel Fast Multipole Algorithm
Ergül, Özgür Salih (2012-02-01)
We present rigorous analysis of double-negative materials (DNMs) with surface integral equations and the multilevel fast multipole algorithm (MLFMA). Accuracy and efficiency of numerical solutions are investigated when DNMs are formulated with two recently developed formulations, i.e., the combined tangential formulation (CTF) and the electric and magnetic current combined-field integral equation (JMCHE). Simulation results on canonical objects are consistent with previous results in the literature on ordin...
Citation Formats
E. Alejaily, “Dynamics for chaos and fractals,” Thesis (Ph.D.) -- Graduate School of Natural and Applied Sciences. Mathematics., Middle East Technical University, 2019.