Domain Structured Dynamics: Unpredictability, chaos, randomness, fractals, differential equations and neural networks

Domain structured dynamics introduces a way for analysis of chaos in fractals, neural networks and random processes. It starts with newly invented abstract similarity sets and maps, which are in the basis of the abstract similarity dynamics. Then a labeling procedure is designed to determine the domain structured dynamics. The results follow the Pythagorean doctrine, considering finite number of indices for the labeling, with potential to become universal in future. The immediate power of the approach for fractals as domains of chaos, revisited famous deterministic and stochastic models, new types of differential equations and neural networks is seen in the book. This is not considered through widening areas, where the notions can be seen and recognized, but by deepening abstraction.


Domain-Structured Chaos in a Hopfield Neural Network
Akhmet, Marat (World Scientific Pub Co Pte Lt, 2019-12-30)
In this paper, we provide a new method for constructing chaotic Hopfield neural networks. Our approach is based on structuring the domain to form a special set through the discrete evolution of the network state variables. In the chaotic regime, the formed set is invariant under the system governing the dynamics of the neural network. The approach can be viewed as an extension of the unimodality technique for one-dimensional map, thereby generating chaos from higher-dimensional systems. We show that the dis...
Implicit monolithic parallel solution algorithm for seismic analysis of dam-reservoir systems
Özmen, Semih; Kurç, Özgür; Department of Civil Engineering (2016)
This research mainly focuses on developing a computationally scalable and efficient solution algorithm that can handle linear dynamic analysis of dam-reservoir interaction problem. Lagrangian fluid finite elements are utilized and compressibility and viscosity of the fluid are taken into consideration during the reservoir modeling. In order to provide computational scalability and efficiency, domain decomposition methods implemented with parallel computing approaches such as Finite Element Tearing and Inter...
Open problems in CEM: A new look at the stability analysis of the finite-difference time-domain method
Ergül, Özgür Salih; Özakın, M. Burak (2014-01-01)
The stability analysis of a numerical time-domain method plays a crucial role in well understanding the numerical behavior of the algorithm. The stability analysis should therefore be investigated in all senses. In this work, a new look at the stability analysis of the Finite-Difference Time-Domain Method is given. A novel link is constructed between the numerical-dispersion analysis and the stability analysis by using the sampled values of the unit space and time steps. Unification of these two analyses th...
Extraneous roots and kinematic analysis of spatial mechanisms and robots
Soylu, Reşit (1997-10-01)
A systematic algorithm, for the complete position analysis of ''any'' single loop spatial mechanism, is introduced. The method yields four non-linear equations, with the least possible total degree. These equations are transformed into algebraic ones, which may be solved, analytically, based upon the concept of resultants. The solutions are free from extraneous roots, since such roots are identified and extracted by the proposed techniques. An algorithm, for the efficient position analysis of mechanisms wit...
Generation of cyclic/toroidal chaos by Hopfield neural networks
Akhmet, Marat (Elsevier BV, 2014-12-05)
We discuss the appearance of cyclic and toroidal chaos in Hopfield neural networks. The theoretical results may strongly relate to investigations of brain activities performed by neurobiologists. As new phenomena, extension of chaos by entrainment of several limit cycles as well as the attraction of cyclic chaos by an equilibrium are discussed. Appropriate simulations that support the theoretical results are depicted. Stabilization of tori in a chaotic attractor is realized not only for neural networks, but...
Citation Formats
M. Akhmet, Domain Structured Dynamics: Unpredictability, chaos, randomness, fractals, differential equations and neural networks. 2021.