Dynamics with Chaos and Fractals

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 heteroclinic motions in economic models.


Dynamics for chaos and fractals
Alejaily, Ejaily; Akhmet, Marat; Department of Mathematics (2019)
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 ...
Chaos in Matrix Gauge Theories with Massive Deformations
Başkan, K.; Kürkcüoğlu, Seçkin; Oktay, O.; Taşcı, Cankut (2022-11-23)
Starting from an SU(N) matrix quantum mechanics model with massive deformation terms and by introducing an ansatz configuration involving fuzzy four- and two-spheres with collective time dependence, we obtain a family of effective Hamiltonians, Hn, (N = 16 (n + 1)(n + 2)(n + 3)) and examine their emerging chaotic dynamics. Through numerical work, we model the variation of the largest Lyapunov exponents as a function of the energy and find that they vary either as ∝ (E − (En)F)1/4 or ∝ E1/4, where (En)F stan...
Algebraic curves hermitian lattices and hypergeometric functions
Zeytin, Ayberk; Önsiper, Mustafa Hurşit; Department of Mathematics (2011)
The aim of this work is to study the interaction between two classical objects of mathematics: the modular group, and the absolute Galois group. The latter acts on the category of finite index subgroups of the modular group. However, it is a task out of reach do understand this action in this generality. We propose a lattice which parametrizes a certain system of ”geometric” elements in this category. This system is setwise invariant under the Galois action, and there is a hope that one can explicitly under...
Hybrid wavelet-neural network models for time series data
Kılıç, Deniz Kenan; Uğur, Ömür; Department of Financial Mathematics (2021-3-3)
The thesis aims to combine wavelet theory with nonlinear models, particularly neural networks, to find an appropriate time series model structure. Data like financial time series are nonstationary, noisy, and chaotic. Therefore using wavelet analysis helps better modeling in the sense of both frequency and time. S&P500 (∧GSPC) and NASDAQ (∧ IXIC) data are divided into several components by using multiresolution analysis (MRA). Subsequently, each part is modeled by using a suitable neural network structure. ...
Mathematical problems of black-box computational technologies for continuum mechanics
Martynenko, Sergey; Zhou, Weixing; Gökalp, İskender; Toktaliev, Pavel; Tarasov, Georgy; Rumiantsev, Egor (2023-02-16)
This paper discusses possible ways of computational technology development for segregated/coupled solving the systems of nonlinear partial differential equations in black-box software. These systems describe physical and chemical processes in the continuum mechanics approximation (multiphysics). The following requirements for the black-box numerical methods are formulated: - robustness (the least number of problem-dependent components); - efficiency (close-to-optimal algorithmic complexity); - parallelism (...
Citation Formats
M. Akhmet, Dynamics with Chaos and Fractals. 2020.