Angela Slavova

Cellular neural networks : dynamics and modelling

Preface. 1: Basic theory about CNNs. 1.1. Introduction to the CNN paradigm. 1.2. Main types of CNN equations. 1.3. Theorems and results on CNN stability. 1.4. Examples. 2: Dynamics of nonlinear and delay CNNs. 2.1. Nonlinear CNNs. 2.2. CNN with delay. 2.3. Examples. 3: Hysteresis and chaos in CNNs. 3.1. CNNs with hystersis in the feedback system. 3.2. Nonlinear CNNs with hysteresis in the output dynamics. 3.3. Feedback and hysteresis. 3.4. Control of chaotic CNNs. 4: CNN modelling in biology, physics and ecology. 4.1. Modelling PDEs via CNNs. 4.2. CNN model of Sine-Gordon equation. 4.3. CNN model of FitzHugh-Nagumo equation. 4.4. CNN model of Fishers equation. 4.5. CNN model of Brusselator equation. 4.6. CNN model of Toda Lattice equation. 4.7. Lotka-Volterra equation and its CNN model. 5: Appendix A: Topological degree method. 6: Appendix B: Hysteresis and its models. 7: Appendix C: Describing function method and its application for analysis of Cellular Neural Networks. References. Index.

CNN model for studying dynamics and travelling wave solutions of FitzHugh-Nagumo equation

In this paper, a cellular neural network (CNN) model of FitzHugh-Nagumo equation is introduced. Dynamical behavior of this model is investigated using harmonic balance method. For the CNN model of FitzHugh-Nagumo equation, propagation of solitary waves have been proved.

Applications of some mathematical methods in the analysis of cellular neural networks

This paper presents a survey of the mathematical tools used for the analysis of cellular neural networks (CNNs). Stability of such dynamical systems is proved by use of Lyapunov function method. Another method for investigation of equilibrium points and their stability is topological degree method. Bifurcations and chaos are studied for cellular neural networks. Approximations of some PDEs via CNN are presented.

Nonlinear waves : an introduction

Compact Traveling Waves, Peakons, Cuspons, Solitons, Kinks and Periodic Solutions of Several Third Order Nonlinear PDE, including Camassa-Holm, Korteweg-De Vries, Burgers Equations and Their Modifications Cellular Neural Network Realization Fluxon and Breathon Solutions of the Sin-Gordon Equation and Their Interaction Stability of Periodic Traveling Wave Solutions for Some Classes of KdV Type Equations Interaction of Peakon Type Solutions of the Camassa-Holm Equation Classical and Continuous Weak Solutions of the Cauchy Problem for the Hunter-Saxton Equation, Peakon Type Solutions Weak Continuous Solutions for the Scalar Conservation Law and Existence Of δ-Shocks Logarithmic Singularities and Microlocal Approach in Studying the Propagation of Nonlinear Waves.

Dynamic properties of cellular neural networks

Dynam}c behav}or of a new class of information-processing systems called Cellular Neural Networks s investigated. In th}s paper we introduce a small parameter n the state equat}on of a cellular neural network and we seek for period}c phenomena. New approach is used for proving stability of a cellular neural network by constructing Lyapunov’s major}z}ng equat}ons. Th}s algorithm }s helpful for find}ng a map from initial continuous state space of a cellular neural network nto d}screte output. A compar}son between cellular neural networks and cellular automata s made.

Dynamics and stability of generalized cellular nonlinear network model

The aim of this paper is the definition of a new model of neural network, called generalized cellular nonlinear network, that covers architectures and dynamics of the well known and widely used classes of feedforward neural networks and cellular neural networks. We show how cellular neural networks and feedforward neural networks can be derived from the general model: moreover we prove a theorem of existence and uniqueness for the solution of the system that describes the generalized cellular nonlinear network dynamics. These results are obtained using the method of Lyapunovs finite majorizing equations that also represents a new approach in studying the stability of cellular neural networks.

Cellular Neural Networks model of risk management

In this paper we shall study Black-Scholes partial differential equation which describes the evolution of the price of asset. We shall apply cellular neural networkspsila approach for studying the model of risk management. Numerical simulations and comparison with the classical results will be presented.

Receptor-Based CNN Model with Hysteresis for Pattern Formation

In this paper receptor-based Cellular Nonlinear Network model with hysteresis is considered. Dynamics and stability of such model are studied by applying describing function technique. Comparison of the obtained results with the classical ones is made as well. Numerical simulations and discussions about the pattern formation in such model are presented

Generalization of CNN with hysteresis nonlinearity

We introduce a general class of neural networks. This new model covers some of the known neural network architectures, including cellular neural networks and Hopfield networks. Hysteresis feedback networks are introduced and compared to the general Hopfield networks in order to prove the existence of hysteresis phenomena in the network.

On the Periodic Solutions in One Dimensional Cellular Nonlinear Networks Based on Josephson Junctions (JJ's)

In this paper we consider an autonomous one dimensional cellular nonlinear network (CNN) that consists of chain of N simple cells based on Josephson Junctions (JJs) coupled by linear inductors. In fact the subject is the well known Josephson Transmission Line (JTL) that is used in many applications in superconductor electronics and more especially in Rapid Single Flux Quantum (RSFQ) technique. When the last cell is connected with the first one a ring of Josephson Junctions is formed. Using the framework of the cellular nonlinear network we consider the JTL under the umbrella of the system theory. Based on describing functions method we rigorously prove the existence and stability of periodic solutions in the cellular nonlinear network model considered. The results are confirmed by simulations